Associative algebra

An associative algebra is an algebra over a field whose multiplication satisfies $(ab)c=a(bc)$. Unless stated otherwise, the algebras considered here have a multiplicative identity.

A primitive ideal of a unital associative algebra is the annihilator of a module for a nonzero simple module. Every maximal proper two-sided ideal is primitive: its quotient is a simple ring, and any simple module over that quotient has zero annihilator there. In a unital Banach algebra, a simple module can be realized as $A/L$ for a maximal left ideal; this ideal is closed, making the representation a bounded algebraically irreducible action on a Banach space.

An Ore extension uses a unital endomorphism $\sigma$ of $A$ and a $\sigma$-derivation $\delta$, satisfying $\delta (ab)=\sigma (a)\delta (b)+\delta (a)b$. Its elements have unique expressions ${\sum }_i{a}_i{t}^i$ and multiplication is determined by the displayed relation. Applying the relation to $t(ab)$ shows why the endomorphism and derivation conditions are exactly the compatibility conditions for associativity. A differential Ore extension has $\sigma =\mathrm{id}$. Adjoining ordinary partial derivatives successively to a polynomial ring gives the ordered-monomial construction of a Weyl algebra.

An algebra involution on a complex associative algebra is a conjugate-linear map satisfying $(xy{)}^{\ast }=y^{\ast }{x}^{\ast }$ and $(x^{\ast }{)}^{\ast }=x$. It fixes the identity when one exists. A star algebra is an algebra equipped with such a map. These algebraic identities do not themselves impose norm continuity or the C-star identity.

An associative algebra with a two-sided multiplicative identity. For a subalgebra of bounded operators, the term unital here means that it contains the identity of the represented Hilbert space.

A complex associative algebra with a conjugate-linear involution satisfying the displayed product-reversal identities. An operator algebra closed under the Hilbert-space adjoint is a star-algebra. A C-star algebra additionally has a complete norm with $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

A Hermitian algebra element of a star-algebra is an element fixed by its algebra involution. The identity and zero are Hermitian algebra elements. For every $x$, both $x{x}^{\ast }$ and $x^{\ast }x$ are Hermitian algebra elements. A real spectrum for all such elements is an extra condition in a general Banach algebra, rather than a consequence of the involution identities alone.

A linear subalgebra closed under the involution. In bounded operators, this means closure under the adjoint operator; neither norm closure nor operator-topology closure is part of the definition. These distinctions matter in the Von Neumann double commutant theorem and the Kaplansky density theorem.

The cocenter of a finite-dimensional algebra is the quotient by the vector subspace spanned by additive commutators $ab-ba$; this subspace need not be an ideal. For a group algebra, its basis is indexed by conjugacy classes: every commutator of basis elements is a difference of conjugates, and every difference of conjugates is a commutator. For $M_n(k)$, off-diagonal matrix units and $E_{ii}-E_{jj}$ show that the quotient is one-dimensional, identified by the trace.

In characteristic $p>0$, the displayed map on the cocenter of a finite-dimensional algebra is additive and Frobenius-semilinear. Expand a $p$th power into words: nonconstant cyclic-word orbits have length $p$ and cancel modulo commutators. Also $(ab{)}^p$ and $(ba{)}^p$ have the same cocenter class, so the map kills additive commutators and is well defined.

Over a perfect field with split semisimple quotient, the stable image of this map has dimension equal to the number of simple modules of the algebra. Indeed, the kernel of $C(A)\to C(A/J(A))$ has representatives in the nilpotent Jacobson radical, so an iterate kills it. On the semisimple algebra quotient, each matrix-block coordinate is a field trace, and $\mathrm{tr}(a^p)=\mathrm{tr}(a{)}^p$. For a split semisimple quotient, Frobenius is bijective on every coordinate. In a group algebra over an algebraically closed field, the stable image has the p-regular element conjugacy classes as its basis, proving the simple-module counting theorem.

For a finitely generated associative algebra, choose matrices for its generators and impose its polynomial relations. This defines an affine variety. With prescribed orthogonal idempotents $1={\sum }_i{e}_i$, require their images to be the standard projections on ${⨁}_i{k}^{n_i}$. The resulting variety is ${\mathrm{Rep}}_A(\mathbf{n})$, acted on by ${\prod }_i{\mathrm{GL}}_{n_i}(k)$. Without a chosen idempotent decomposition, the usual variety is ${\mathrm{Rep}}_A(r)={\mathrm{Hom}}_{k\text{-alg}}(A,M_r(k))$. Its orbits are module isomorphism classes, and their stabilizers are automorphism groups.

The unit group $E^{\times }$ is a nonempty open subset of the affine space $E$, detected by the determinant of left multiplication. It is therefore an irreducible connected linear algebraic group. Its map to $(E/J(E){)}^{\times }$ has kernel $1+J(E)$, a closed unipotent subgroup since the radical is nilpotent.

The free associative algebra has a basis of finite words in its generators, including the empty word as identity. Multiplication concatenates words without commutativity relations. It is the path algebra of the quiver with r loops. With one generator it is a polynomial algebra.

A finite-dimensional unital associative algebra over a field admitting a linear functional $\epsilon$ whose pairing $(a,b)\mapsto \epsilon (ab)$ is nondegenerate. The dual numbers with functional extracting the nilpotent coefficient are an example; semisimplicity is unnecessary.

The same underlying module as an associative algebra, with multiplication $a{\cdot }_{\mathrm{op}}b=ba$. Right multiplications identify $H^{\mathrm{op}}$ with ${\mathrm{End}}_H(H)$ for the left regular module.

A right ideal $I$ in a unital associative algebra $A$ is a vector subspace with $IA\subseteq I$, equivalently a submodule of the right regular module. It need not be a left ideal. Its finite generation refers to right multiplication by elements of $A$.

A proper right ideal $L$ is maximal if no proper right ideal strictly contains it. Equivalently, $A/L$ is a simple module. The Jacobson radical is the intersection of all maximal right ideals.

A formal associative deformation is a unital $k[[t]]$-bilinear product continuous for the adic topology on $A[[t]]$ of the form $a\ast b=ab+{\sum }_{r\ge 1}{t}^r{\mu }_r(a,b)$. Each coefficient ${\mu }_r$ is a $k$-bilinear map $A\otimes A\to A$, and the whole product satisfies associativity. For an infinite-dimensional algebra, $A[[t]]$ is the completed tensor product ${\underleftarrow{\lim }}_{n}A\otimes k[t]/(t^n)$ and generally differs from the ordinary $A\otimes k[[t]]$.

The algebra over $k[t]$ defined by $yx-xy=t$ has ordered basis $x^i{y}^j$, by its differential Ore-extension construction. Its $t$-adic completion is a formal associative formal associative deformation of $k[x,y]$. In characteristic zero its first-order multiplication term is $f(a,b)={\partial }_{y}a{\partial }_{x}b$. The cocycle is not a coboundary: a degree-two coboundary on a commutative algebra is symmetric, whereas $f(y,x)-f(x,y)=1$.

The order-$\epsilon$ part of associativity is $af(b,c)-f(ab,c)+f(a,bc)-f(a,b)c=0$, so $f$ is a Hochschild cocycle. A change of coordinates $T=1+\epsilon g$ changes $f$ to $f+\delta g$. Therefore equivalence classes of first-order associative deformations of a fixed underlying algebra are its second Hochschild cohomology. Unital deformations use normalized cochains.

Expanding associativity for $a\ast b=ab+tf(a,b)+t^2{\mu }_2(a,b)$ modulo $t^3$ gives the displayed equation. If $\delta f=0$, its right side is a degree-three Hochschild cocycle. Its class in $H{H}^3(R,R)$ must vanish to extend the first-order formal associative deformation to second order. A nonzero class in $H{H}^2$ therefore need not integrate to a full formal associative deformation.

A formal associative deformation is trivial if a $k[[t]]$-linear automorphism continuous for the adic topology $T=\mathrm{id}(\mathrm{mod}t)$ fixing the identity transports the deformed multiplication to the original one. This is equivalence by a formal change of coordinates, stronger than merely an abstract algebra isomorphism between unspecified middle terms.

If $H{H}^2(A,A)=0$, every formal associative deformation on the adic completion of a module $A[[t]]$ is trivial. After killing lower coefficients, associativity makes the order-$r$ coefficient a Hochschild cocycle ${\mu }_r=\delta g_r$. Transport by $\mathrm{id}+t^r{g}_r$ subtracts $\delta g_r$ and kills it. The successive transformations stabilize modulo each $t^n$ and converge to an invertible change of coordinates. Completeness, and equivalence congruent to the identity modulo $t$, are essential to this argument.

For a nonzero finitely generated algebra over a field, choose a finite-dimensional generating vector subspace $V$ containing $1$. The dimension is $\mathrm{GKdim}A={\mathrm{limsup}}_{n\to \infty }\log ({\dim }_k{V}^n)/\log n$. Changing $V$ only rescales the filtration index by bounded factors. A nonzero finite-dimensional algebra has dimension zero; a polynomial ring in $r$ variables has dimension $r$. The zero algebra is often assigned dimension $-\infty$.

For a nonzero right module $M$ that is a finitely generated module, choose a finite-dimensional generating vector subspace $W$ and define ${\mathrm{GKdim}}_{A}M={\mathrm{limsup}}_{n\to \infty }\log ({\dim }_{k}W{V}^n)/\log n$, where $V$ generates the algebra and contains $1$. For a left module use $V^{n}W$. It is independent of these choices and measures polynomial-order growth of a generating filtration of a module.

For a nonzero finitely generated Weyl algebra module, choose a finite-dimensional generating subspace $M_0$. Under the Bernstein filtration, $\dim (F_j{A}_n{M}_0)$ is eventually polynomial because the graded module is finite over a polynomial ring. Its degree is independent of the generating subspace, since two generating filtrations bound each other after fixed shifts. It equals the Gelfand–Kirillov dimension of a module.

A good Bernstein filtration on a finitely generated module produces a finite graded module over $\mathrm{gr}{A}_n\cong \mathbb{C}[x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n]$. Its support is the characteristic variety for this filtration. Its algebraic dimension is the Bernstein growth dimension of a Weyl algebra module, which is independent of the choice of good filtration.

Fix the Bernstein filtration on the Weyl algebra. Two good filtrations of a module are bounded shifts of one another: $F_{i}M\subseteq G_{i+c}M$ and conversely. If an operator of degree $d$ has symbol annihilating ${\mathrm{gr}}_{F}M$, its $r$th power lowers the expected filtration degree by $r$. For $r>2c$, comparison of filtrations forces the same power of its symbol to annihilate ${\mathrm{gr}}_{G}M$. Reversing the roles gives equality of the radical annihilator ideals and hence equality of support of a module. The algebra filtration is held fixed; changing between order and Bernstein filtrations need not preserve the same variety.

A nonzero finitely generated Weyl algebra module over $A_n(\mathbb{C})$ is holonomic when it attains the lower bound in the Bernstein inequality for Weyl algebra modules. The zero module is also conventionally called holonomic. Nonzero subquotients of a holonomic module remain holonomic. Additivity and integrality of the leading Hilbert-polynomial multiplicity then bound the number of nonzero factors in any submodule chain, giving finite composition length.

Every nonzero subquotient of a holonomic Weyl algebra module over a characteristic-zero field has dimension $n$, by Bernstein inequality for Weyl algebra modules and dimension and multiplicity in a filtered exact sequence. Each strict step in a submodule chain consumes positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral. There are therefore at most $m(M)$ strict steps. The module is Artinian and Noetherian, and its module length is bounded by that multiplicity.

For a nonzero finitely generated module over the complex Weyl algebra $A_n$, its Bernstein growth dimension of a Weyl algebra module is at least $n$. The faithful finite-step action of a Weyl algebra gives $\dim F_j{A}_n\le (\dim M_j)(\dim M_{2j})$. Comparing degrees $2n$ and $2d(M)$ proves the inequality. The polynomial module has dimension $n$, so the bound is sharp. This is an algebraic growth inequality, distinct from probabilistic Bernstein inequalities.

For $M_j=F_j{A}_n{M}_0$ and $M_0\ne 0$, the action map in the display is injective. If $a\in F_j$ kills $M_j$, each commutator $[a,z]$ with a generator lies in $F_{j-1}$ and kills $M_{j-1}$. Induction makes all these commutators zero, so $a$ is scalar; since it kills $M_0$, it is zero. This finite-step faithfulness does not require the whole module to be finite-dimensional.

For a nonzero element of a Weyl algebra over a characteristic-zero field, choose a monomial $x^{\alpha }{D}^{\beta }$ of maximal total degree with coefficient $c_{\alpha \beta }\ne 0$. The displayed iterated commutators kill every lower-degree monomial. Another monomial of the same degree survives only if all its exponents dominate $(\alpha ,\beta )$, which forces equality. Hence the result is a nonzero scalar. If the element has filtered degree at most $j$, expansion of these commutators gives a sum of terms $uav$ with both $u$ and $v$ of degree at most $j$. This proves faithful finite-step action of a Weyl algebra over any characteristic-zero field.

A block of an associative algebra with a finite decomposition of its identity is a two-sided direct summand $Ae$ corresponding to a primitive central idempotent $e$. Its identity is $e$, and it cannot be split into two nonzero algebras by central idempotents. For a group algebra of a finite group, this agrees with a block of a group algebra. Orthogonal primitive central idempotents give $A={⨁}_{i}A{e}_i$.

A commutative unital subalgebra $D$ of an associative algebra $A$ is maximal commutative if no larger commutative subalgebra contains it. Equivalently $C_A(D)=D$: any element of its centralizer of a subalgebra generates a commutative algebra together with $D$. The full diagonal matrix algebra inside $M_n(\mathbb{C})$ is an example.

For a subalgebra $B$ of an associative algebra $A$, its centralizer isIt is a subalgebra of $A$. When $A$ is a group algebra and $B$ is the group algebra of a subgroup $H$, its elements are exactly the linear combinations with coefficients constant on the conjugation orbits of $H$.

The commutant of $A\subseteq \mathrm{End}(T)$ is $A\prime =\{b:ba=ab\text{ for all }a\in A\}$. This centralizer is an operator algebra. The double-centralizer theorem for semisimple operator algebras and Schur–Weyl duality use this meaning, which is distinct from the centralizer of a single element of a group.

For a unital star-subalgebra $A$ of the bounded operators on a complex Hilbert space, its strong operator topology and weak operator topology closures equal its double commutant of an operator algebra. To approximate $T\in A^{\prime \prime }$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A^{\prime }$, so it commutes with the diagonal action of $T$. Therefore the image tuple under $T$ lies in that cyclic subspace and can be approximated by a single $a\in A$.

For a unital star-subalgebra $A\subseteq B(H)$, its unit ball is dense in the unit ball of $A^{\prime \prime }$ in the strong operator topology. The same holds for self-adjoint unit balls and for positive unit balls, and the full unit balls may be approximated strongly together with their adjoints. Passing first to the norm closure allows continuous functional calculus; resolvent identities prove strong continuity of continuous functions vanishing at infinity, even for unbounded approximating nets of self-adjoint operators. A cutoff function supplies the norm bound, and a two-by-two self-adjoint matrix removes the self-adjoint restriction.

A unital subalgebra $A\subseteq \mathrm{End}(V)$ closed under the adjoint operator, on a finite-dimensional real or complex inner product space, equals its double commutant of an operator algebra. Here is a proof that does not require a classification of algebras. For $N=\dim V$, choose a basis $(v_1,\dots ,v_N)$ and let $W=\{(a{v}_1,\dots ,a{v}_N):a\in A\}\subseteq V^N$. The diagonal action of $A$ preserves $W$ and $W^{\perp }$, so the orthogonal projection onto $W$ commutes with that action. Each matrix entry of this projection belongs to $A^{\prime }$. Hence the diagonal action of $b\in A^{\prime \prime }$ commutes with the projection and preserves $W$. Since $1\in A$, the tuple $(v_1,\dots ,v_N)$ belongs to $W$, and therefore $(b{v}_1,\dots ,b{v}_N)=(a{v}_1,\dots ,a{v}_N)$ for one $a\in A$. This implies $b=a$. Without the unital hypothesis the conclusion is $A^{\prime \prime }=A+\mathbb{F}1$.

For a finite-dimensional semisimple algebra acting faithfully on $T$, write $T={⨁}_i{S}_i\otimes M_i$. Then the algebra acts by the full matrix algebra on each simple factor, while its commutant acts by the full matrix algebra on each multiplicity factor. Taking the commutant twice recovers the original image. This is the algebraic basis of Schur–Weyl duality.

For an associative algebra $A$, its matrix algebra $M_n(A)$ consists of all $n$ by $n$ matrices over $A$, with ordinary matrix multiplication. If $A=k$ is a field, this is ${\mathrm{End}}_k(k^n)$ and is a central simple algebra. Full matrix algebras over division rings are the factors in the Artin–Wedderburn theorem.

A block of a finite-dimensional associative algebra $R$ is a two-sided ideal $Re$ associated with a primitive idempotent $e$ in the center of an associative algebra. Its identity is $e$, and $R$ is the product of these blocks. An $R$-module belongs to this block if $eM=M$. For a semisimple algebra, the Artin–Wedderburn theorem says the blocks are its full matrix algebras over division rings. For a group algebra this specializes to a block of a group algebra.

For a finite-dimensional associative algebra, form a graph on its simple modules, joining $S,T$ when ${\mathrm{Ext}}^1(S,T)$ or ${\mathrm{Ext}}^1(T,S)$ is nonzero. Its connected components are exactly the simple modules belonging to each block of a finite-dimensional algebra. They are also the components generated by sharing a Jordan–Hölder factor occurrence in an indecomposable representation that is a projective module.

Indeed, Ext separation of finite-length modules would split the regular module of a block into canonical summands if that block had two graph components. Right multiplication preserves these summands, so its projection supplies a nontrivial central idempotent, contradicting the definition of a block. An indecomposable projective belongs to one block. Finally, a nonsplit extension $0\to U\to V\to W\to 0$ of simple modules is a quotient of the projective cover of $W$: a lift onto $W$ must contain $U$, since otherwise the extension splits. Thus its two endpoints share factors in an indecomposable projective.

A finite-dimensional algebra over an algebraically closed field is basic when its semisimple quotient is a finite product of copies of the field, equivalently when every simple module occurs with multiplicity one in the semisimple quotient of the regular representation.

For a finite-dimensional associative algebra over an algebraically closed field, the Ext quiver has a vertex for each simple module isomorphism class and $a_{ij}$ arrows $i\to j$. For a basic algebra with $P_i=A{e}_i$, applying ${\mathrm{Hom}}_A(-,S_j)$ to the projective-cover sequence gives $a_{ij}=\dim e_j(J/J^2)e_i$. Our path algebra convention multiplies consecutive arrows $\alpha :i\to j$, $\beta :j\to l$ as $\beta \alpha$, making left modules have arrow maps $V_i\to V_j$. The orientation convention must be stated when comparing other definitions.

A central simple algebra over a field $k$ is a finite-dimensional associative $k$-algebra that is a simple ring and whose center is exactly $k$.

For a field of characteristic other than two and nonzero $a,b$, the quaternion algebra has basis $1,i,j,ij$ with the displayed relations. It is a four-dimensional central simple algebra. After adjoining a square root of $a$, diagonalizing $i$ and taking $j=\left(\begin{array}{cc}0& b\\ 1& 0\end{array}\right)$ identifies it with $M_2$. Quaternion conjugation fixes scalars and negates $i,j,ij$, giving norm $x_0^2-a{x}_1^2-b{x}_2^2+ab{x}_3^2$. An element is invertible exactly when its norm is nonzero. Thus the quaternion algebra is either a division algebra or a split matrix algebra. The familiar quaternion algebra is $(-1,-1{)}_{\mathbb{R}}$.

For $p\equiv 3(\mathrm{mod}4)$ and $z=x+iy\ne 0$ with rational $x,y$, divide $x^2+y^2$ by the square of the common minimum $p$-adic valuation. Its remaining unit cannot vanish modulo $p$, because $-1$ is not a square there. The norm therefore has even valuation. In particular a rational parameter $a$ with odd valuation at such a prime is not a norm, and $(-1,a{)}_{\mathbb{Q}}$ is a division quaternion algebra: its norm $N(z)-aN(w)$ cannot vanish for a nonzero element $z+wj$.

The reduced norm of an element of a central simple algebra of degree $d$ is its determinant after any splitting isomorphism to $M_d$. Inner automorphisms of a matrix algebra preserve that determinant, while Galois invariance descends its polynomial coefficients to the center. It is a multiplicative homogeneous degree-$d$ polynomial, vanishing exactly on noninvertible elements. On a central division algebra it is anisotropic.

Every $L$-algebra automorphism of $M_d(L)$ is conjugation by an invertible matrix. For matrix units, the vectors $h(E_{i1})w$ with $0\ne w\in \mathrm{im}h(E_{11})$ form a basis in which $h$ acts in the usual way on all matrix units. Scalar conjugating matrices act trivially, giving ${\mathrm{Aut}}_L(M_d(L))={\mathrm{PGL}}_d(L)$.

An extension $E/K$ splits a central simple algebra $A$ of degree $d$ when $A{\otimes }_{K}E\cong M_d(E)$. A finite separable splitting field exists. Splitting is preserved by further scalar extension and is equivalent to the vanishing of the restricted Brauer group class.

A central simple algebra $A$ has dimension $d^2$ over its center. Its degree is this integer $d$. Over a splitting field it becomes $M_d$, so degree records the matrix size independently of the chosen splitting.

Every finite-dimensional semisimple algebra is a finite product of matrix algebras over division algebras. In particular, every central simple algebra over $k$ is isomorphic to $M_r(D)$ for a finite-dimensional central division algebra $D$ over $k$. The integer $r$ and the isomorphism class of $D$ are unique.

Indeed, decompose the right regular module of a finite-dimensional semisimple algebra $A$ aswhere the $S_i$ are pairwise nonisomorphic simple modules. The Schur lemma makes $D_i={\mathrm{End}}_A(S_i)$ a division ring and gives ${\mathrm{Hom}}_A(S_i,S_j)=0$ for $i\ne j$. Left multiplication identifiesConversely, the column module over $M_n(D)$ is simple and the regular module is a direct sum of $n$ copies of it, so finite products of such matrix rings are semisimple. The simple-module decomposition also proves uniqueness up to permuting the factors.

For a nonzero finite-dimensional complex vector space $V$, a unital matrix algebra acting irreducibly on $V$ is the full endomorphism algebra. Its Jacobson radical annihilates $V$: otherwise the radical times $V$ is all of $V$, contradicting nilpotence. Faithfulness kills the radical, and the Artin–Wedderburn theorem then leaves one simple matrix factor acting on its standard column module. This is a matrix-algebra theorem, distinct from the orbit-counting Burnside lemma and the finite-group Burnside's theorem.

The tensor product over $k$ of two central simple $k$-algebras is central simple. After extending scalars to an algebraic closure, both factors and their tensor product become full matrix algebras; faithful flatness then descends simplicity and the center.

The Brauer group of a field $k$ consists of Morita equivalence classes of central simple $k$-algebras. Multiplication is induced by tensor product, the identity is $[k]$, and the inverse of $[A]$ is the class of the opposite algebra $A^{\mathrm{op}}$.

The Brauer class of a central simple algebra is its equivalence class under adjoining matrix-algebra factors. Two algebras have the same class exactly when their central division-algebra representatives are isomorphic. The zero class consists of split matrix algebras; the inverse of $[A]$ is the class of the opposite algebra. For a cyclic algebra $(L/K,\sigma ,a)$, the class is zero exactly when $a$ is a field norm from $L$.

For a number field $K$, localization of the Brauer group is injective. Its image is exactly the finite-support tuples of local classes whose local Brauer invariants sum to zero. The non-Archimedean summands have invariant group $\mathbb{Q}/\mathbb{Z}$, real summands have $\{0,1/2\}$, and complex summands are zero. For a cyclic character $\chi$ and parameter $a$, the invariant of the local cyclic algebra is $\chi ({\mathrm{Art}}_{K_v}(a))$. Thus the zero-sum condition on principal parameters is the additive form of Artin reciprocity.

Associate to each prime $p\equiv 3(\mathrm{mod}4)$ the class of $(-1,p{)}_{\mathbb{Q}}$. These classes have order two, since quaternion conjugation identifies the algebra with its opposite. The product rule $[(-1,a)][(-1,b)]=[(-1,ab)]$ follows by factoring their tensor product into the commuting subalgebras generated by $i_1,j_1{j}_2$ and by $i_1{i}_2,j_2$; the second is the split algebra $(1,b)$. Every nonempty finite product of distinct such primes has an odd valuation at one of them. The quaternion norm obstruction at primes congruent to three modulo four therefore makes its class nonzero, proving independence. Euclid's construction $4p_1\cdots p_m-1$ proves there are infinitely many such primes.

In characteristic $p$, the exact sequence $1\to K_s^{\times }\stackrel{p}{\to }{K}_s^{\times }\to K_s^{\times }/(K_s^{\times }{)}^p\to 1$ has a quotient killed by $p$. The characteristic p Galois dimension bound makes its second cohomology zero. Hence multiplication by $p$ on $\mathrm{Br}(K)$ is surjective. The root map is injective, and its failure of surjectivity on a separable closure is encoded by the quotient.

For a complete discretely valued field $K$ with perfect residue field $k$, there is a split exact sequence $0\to \mathrm{Br}(k)\to \mathrm{Br}(K)\stackrel{{\partial }_K}{\to }{H}^1(k,\mathbb{Q}/\mathbb{Z})\to 0$. The residue is canonical. A uniformizer chooses a splitting $\chi \mapsto (\chi ,\pi )$; this choice does not make the residue itself noncanonical.

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.

For a finite Galois extension $L/K$ of non-Archimedean local fields, its relative Brauer group is the kernel of restriction $\mathrm{Br}(K)\to \mathrm{Br}(L)$ and identifies with $H^2(\mathrm{Gal}(L/K),L^{\times })$. Restriction multiplies the local Brauer invariant by $[L:K]$. This kernel is therefore cyclic of order $[L:K]$, and the indicated class is its distinguished generator. In an unramified extension with arithmetic Frobenius generator it is represented by the cyclic algebra with parameter a uniformizer.

For a finite extension of local fields $L/K$, restriction multiplies the local Brauer invariant by $[L:K]$ and corestriction preserves it. For restriction, the residue-character value multiplies by the residue degree and the parameter valuation by the ramification index. For corestriction, lift the residue character using divisibility of $\mathbb{Q}/\mathbb{Z}$ and use the norm projection formula for cyclic Brauer pairings.

The period is the order of a class in the Brauer group. It divides its index. Over a local field with finite residue field, the period and index are both the reduced denominator of its local Brauer invariant.

If $A\cong M_r(D)$ with $D$ a central division algebra, its index is $\mathrm{deg}D$. It depends only on the Brauer group class. The index divides every finite splitting-field degree and equals the degree of a maximal subfield of the division representative.

For a finite Galois extension $L/k$, Galois group $G$, and normalized two-cocycle $\varphi :G\times G\to L^{\times }$, the crossed-product algebra iswith $u_{\sigma }a=\sigma (a)u_{\sigma }$ and $u_{\sigma }{u}_{\tau }=\varphi (\sigma ,\tau )u_{\sigma \tau }$. It is a central simple $k$-algebra split by $L$.

For a cyclic extension $E/K$ of degree $m$ with generator $\sigma$ and $a\in K^{\times }$, the cyclic algebra is generated by $E$ and $u$ with $ub=\sigma (b)u$, $u^m=a$. It is a central simple algebra of degree $m$. Its Brauer class can also be written $(\chi ,a)$, where $\chi (\sigma )=1/m$.

If $a=N_{L/K}c$, let $L$ act on itself by multiplication and let the extra generator act by $c\sigma$. Its $n$th power is multiplication by $a$, and these operators give the cyclic algebra as ${\mathrm{End}}_K(L)$. Conversely, if the algebra is a matrix algebra on an $n$-dimensional vector space, its embedded degree-$n$ field $L$ makes that space one-dimensional over $L$. The extra generator is then $c\sigma$ in an $L$-basis, so its $n$th power forces $a=Nc$.

For a finite extension $L/K$, an unramified or general cyclic character $\chi$ over $K$, and $a\in L^{\times }$, transfer satisfies ${\mathrm{Cor}}_{L/K}(\mathrm{Res}\chi ,a)=(\chi ,N_{L/K}a)$. It is the cup-product projection formula combined with the field norm on multiplicative degree-zero coefficients. With the norm definition of Brauer transfer it also covers purely inseparable steps.

Cohomologous normalized two-cocycles define isomorphic crossed-product algebras: replacing $\varphi$ by the coboundary associated with a one-cochain rescales the basis elements $u_{\sigma }$. Consequently the crossed-product construction induces a map

The center of an associative algebra $A$ is

A central idempotent is an idempotent in the center of an associative algebra. It gives a decomposition $A=Ae\oplus A(1-e)$ into two-sided ideals, with products between the summands zero. The identities of these summands are $e$ and $1-e$.

A nonzero central idempotent is primitive central if it is not a sum of two nonzero orthogonal central idempotents. For an Artinian ring, it determines a block of an Artinian algebra. This differs from a primitive idempotent in the whole ring: for a matrix algebra of size greater than one over a field, $1$ is primitive central but is a sum of diagonal idempotents.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

The matrix-block sizes in the Artin–Wedderburn theorem for $\mathbb{C}[G]$ recover the irreducible character degrees with their multiplicities. They determine the degree-square sum and the number $|G:G^{\prime }|$ of linear characters. Isaacs' character-degree criterion for p-nilpotency therefore detects each normal p-complement and applying it for every prime also detects nilpotent groups. The set of distinct degrees without multiplicities does not contain the same information.

Write the group algebra as $\mathbb{C}[X,X^{-1}]\oplus \mathbb{C}[X,X^{-1}]Y$, with $Y^2=1$ and $YX=X^{-1}Y$. Commuting $f(X)+g(X)Y$ with $X$ forces $g=0$, and commuting with $Y$ forces $f(X)=f(X^{-1})$. Every symmetric Laurent polynomial is a polynomial in $z=X+X^{-1}$, using the recursion $X^{m+1}+X^{-m-1}=z(X^m+X^{-m})-(X^{m-1}+X^{1-m})$. The algebra is free of rank four over this center, with basis $1,X,Y,XY$; the relation $X^2-zX+1=0$ proves spanning, and applying $X\mapsto X^{-1}$ proves independence of $1,X$.

If the order of a finite abelian group $\Delta$ is invertible in the coefficient ring and all its character values belong to that ring, its character idempotents are orthogonal and sum to one. They decompose any module into character components. In a cyclotomic tower with odd $p$, the Teichmüller character provides these components integrally over ${\mathbb{Z}}_p$.

For a conjugacy class $C$ of a finite group, $z_C={\sum }_{g\in C}g$ is central in its group algebra. In a complex irreducible representation of degree $d$ and character of a representation $\chi$, it acts by $|C|\chi (g)/d$; its trace gives this formula and Schur lemma gives scalarity.

The group norm element in the group algebra $kG$ is $N_G={\sum }_{g\in G}g$. It spans the invariant subspace of the left regular group representation, and $N_G^2=|G|N_G$. If the characteristic divides $|G|$, a projection from $kG$ onto $k{N}_G$ would send $N_G$ to both itself and zero, proving that $kG$ is not a semisimple algebra. Its action on indecomposable modules is described by group norm element detects the trivial projective cover.

For a profinite group $G$ and a finite or profinite coefficient ring $R$, the completed group algebra iswhere $U$ ranges over the open normal subgroups. It records convergent noncommutative power series in topological generators of $G$.

Choose ordered minimal topological generators $a_1,\dots ,a_d$ of a uniform pro-p group for odd $p$, put $b_i=a_i-1$, and choose $1/(p-1)<s<1$. Ordered monomials $b^{\alpha }=b_1^{{\alpha }_1}\cdots b_d^{{\alpha }_d}$ give power-series coordinates in the integral completed group algebra. The displayed Non-Archimedean absolute value filtration gives a complete normed ${\mathbb{Q}}_p$-algebra after completion of the rational group algebra. The power relation raises valuation by one, and interchanging $b_i,b_j$ adds terms of valuation at least $1+s>2s$, so multiplication does not decrease total valuation. The p-adic logarithm and p-adic exponential are inverse on the relevant small balls. In particular $\log G$ is a faithful realization of the Lie algebra of a uniform pro-p group, useful for verifying its intrinsic limit operations and the Baker--Campbell--Hausdorff formula without assuming the correspondence in advance.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A p-adic measure gives the compatible finite-quotient elements ${\sum }_{gU}\mu (gU)[gU]\in I[G/U]$. The transition maps sum coefficients over fibers. Conversely compatible coefficients define values on cosets, then all clopen sets, and hence integration. Thus the inverse limit Iwasawa algebra is the ring of measures, with multiplication given by convolution of p-adic measures. Completeness of $I$ suffices even when its residue field is infinite.

Choose a topological generator $\gamma$ of the Galois group of a Zp-extension and set $T=\gamma -1$. The completed group algebra becomes the formal power series ring ${\mathbb{Z}}_p[[T]]$, a two-dimensional complete regular local integral domain with maximal ideal $(p,T)$. Its finite-layer quotient is $\Lambda /((1+T{)}^{p^n}-1)\cong {\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$.

A homomorphism between finitely generated one-variable Iwasawa modules is a pseudo-isomorphism if its kernel and cokernel are finite. Finite errors do not change the Iwasawa-module rank or the characteristic ideal of a torsion module. They can still change its integral structure, so a pseudo-isomorphism is weaker than an isomorphism.