Ring

A ring has addition and multiplication, with an abelian group under addition and multiplication distributive over addition.

A subset closed under addition, additive inverses and multiplication, with the induced operations. Under the unital convention used here it also contains the identity of $R$. A subring of an integral domain is again an integral domain. The ring of invariants is one example: fixed elements remain fixed under all ring operations.

A topological ring is a ring equipped with a topology making addition, additive inversion and multiplication continuous functions. An adic topology is obtained from powers of an ideal.

For a two-sided ideal $I$ of a ring $R$, the powers $I^n$ form a neighborhood base at zero for the adic topology. For an $R$-module $M$, use $I^{n}M$ on the left or $M{I}^n$ on the right. Convergence means that for every $n$ the differences eventually belong to the corresponding submodule. The formal power series ring $k[[t]]$ and formal power series module $M[[t]]$ are complete for $I=(t)$; convergence fixes each finite list of coefficients eventually. This is formal convergence, without an analytic condition on the size of coefficients.

The adic completion is the inverse limit of the quotient modules $M/I^{n}M$. An element is a compatible family of residue classes at every order. For $R=k[t]$ and $M=V{\otimes }_{k}k[t]$, this gives the formal power series module $V[[t]]$. The canonical map from $M$ need not be injective in general: its kernel is ${\bigcap }_n{I}^{n}M$. It is injective for this polynomial example because a nonzero polynomial has finite degree. Successive changes of coordinates in formal rigidity from vanishing second Hochschild cohomology converge in this completion because the order-$r$ change is the identity modulo $t^r$.

A ring is von Neumann regular if every element $a$ has an element $b$ with $a=aba$. In a commutative ring, this is $a=a^{2}b$. An arbitrary direct product of rings whose factors are fields has this property by taking inverses in each nonzero coordinate. Every localization at a prime ideal of a commutative von Neumann regular ring is a field.

The direct product of a family of rings consists of all tuples of their elements, with coordinatewise addition and multiplication. Unlike a direct sum, tuples may have infinitely many nonzero coordinates.

The characteristic of a ring $R$ is the least positive integer $n$ satisfying $n\cdot 1_R=0$, or zero if no such integer exists. A nonzero ring of positive characteristic has prime characteristic when it is an integral domain.

A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse. Its multiplication need not be commutative.

A nonzero ring is simple when its only two-sided ideals are zero and the whole ring.

A semisimple ring is a ring that is semisimple as a module over itself. Equivalently, every module over it is semisimple, so every short exact sequence of modules splits.

The group ring $R[G]$ consists of finite formal sums ${\sum }_{g\in G}{r}_{g}g$ with coefficients in a ring $R$. Addition is coefficientwise and multiplication extends the group operation and multiplication in $R$ distributively. A group algebra is the case in which $R$ is a field.

The augmentation homomorphism $\epsilon :R[G]\to R$ sends ${\sum }_g{r}_{g}g$ to ${\sum }_g{r}_g$. Its kernel is the augmentation ideal.

The group ring of an infinite cyclic group is the Laurent polynomial ring $R=\mathbb{Z}[t,t^{-1}]$, with augmentation $\epsilon (t)=1$. Multiply any element in the kernel by a power of $t$ to obtain a polynomial vanishing at one. The factor theorem makes it divisible by $t-1$. Thus the augmentation ideal is $(t-1)R$. Since $R$ is an integral domain, multiplication by $t-1$ is an injective module map, so this ideal is a free $R$-module of rank one, even though its additive group has infinite rank.

A noncommutative ring is a ring whose multiplication is not required to be commutative.

An element $x$ of a ring is nilpotent when $x^n=0$ for some positive integer $n$. A field has no nonzero nilpotent elements.

A commutative ring is a ring whose multiplication satisfies $ab=ba$.

A regular sequence in a commutative ring $A$ is a finite list such that $(f_1,\dots ,f_r)$ is proper and $f_j$ is a non-zero-divisor on $A/(f_1,\dots ,f_{j-1})$ for each $j$. In a graded ring with homogeneous $f_j$, injectivity can be checked on homogeneous elements because components of distinct degrees cannot cancel. It gives short exact sequences $0\to R_{j-1}(-\mathrm{deg}{f}_j)\to R_{j-1}\to R_j\to 0$ for the successive graded quotients.

The dual-number ring over a field $k$ consists of elements $a+b\epsilon$ with ${\epsilon }^2=0$. The element $\epsilon$ is a nonzero nilpotent element; its ideal is the ring's only prime ideal. Hence its affine scheme has the same one-point topology as $\mathrm{Spec}k$ while its structure sheaf differs.

The formal power series ring $R[[T]]$ consists of all expressions ${\sum }_{n\ge 0}{a}_n{T}^n$ with $a_n\in R$. Addition is coefficientwise, while multiplication uses the Cauchy product; no analytic convergence is required.

A formal power series ring is a reduced ring if and only if its coefficient ring is reduced. A nonzero constant nilpotent proves one direction. In the other direction, a nonzero series with first nonzero coefficient $a_r$ has first coefficient $a_r^m$ in its $m$th power; this is nonzero over a reduced ring.

For a vector space $M$ over a field $k$, its formal power series module is $M[[t]]={\underleftarrow{\lim }}_{n}M{\otimes }_{k}k[t]/(t^n)$. It is complete for the filtration by $t^{n}M[[t]]$. The ordinary tensor product $M{\otimes }_{k}k[[t]]$ embeds in it and consists of series whose coefficients span a finite-dimensional vector subspace; it equals $M[[t]]$ if $M$ is finite-dimensional. The inclusion is proper for infinite-dimensional $M$, as shown by a series with linearly independent coefficients.

A formal power series ring $R[[x]]$ is Noetherian if and only if $R$ is Noetherian. In the forward direction, use the quotient ring $R[[x]]/(x)\cong R$. In the reverse direction, for an ideal $I$ consider the ascending sequence of ideals of $R$ formed by the coefficient of $x^n$ in $I\cap x^{n}R[[x]]$. Stabilization and finite generation give a finite list of series that generate $I$ by successive coefficient cancellation.

For an ideal $H\subseteq R[[X]]$, let $A_n$ consist of coefficients of $X^n$ in members of $H$ with all lower coefficients zero. These are ideals of $R$ and $A_n\subseteq A_{n+1}$, by multiplication by $X$. If $R$ is Noetherian, this chain stabilizes. Choose series whose leading coefficients generate the finitely many distinct coefficient ideals, then cancel coefficients successively. The accumulated multipliers are formal power series, giving an ordinary finite ideal generating set for $H$, rather than merely a dense subideal.

The ordinary generating function of a sequence $(a_n{)}_{n\ge 0}$ is the formal power series $A(z)={\sum }_{n\ge 0}{a}_n{z}^n$. Algebraic identities for $A$ encode recurrences and counting constructions for the coefficients.

A reduced ring is a commutative ring with no nonzero nilpotent element. Equivalently, its nilradical is zero.

A graded ring is a direct sum $R={⨁}_{n\ge 0}{R}_n$ with $R_m{R}_n\subseteq R_{m+n}$.

A graded module over a graded ring $R={⨁}_i{R}_i$ is an $R$-module with a direct sum decomposition $M={⨁}_d{M}_d$ satisfying $R_i{M}_d\subseteq M_{i+d}$. A graded homomorphism preserves these degrees. Over a polynomial ring with positive variable degrees, a finitely generated module of this kind is bounded below and has finite-dimensional graded components over the coefficient field.

If $f$ is homogeneous of degree one, multiplication by powers of the invertible element $f$ identifies every homogeneous component of $M_f$ with $(M_f{)}_0$. A homogeneous element $m$ of degree $a$ is written $f^a(f^{-a}m)$. Thus the displayed map is an isomorphism of graded modules. In particular, the degree-zero part of a tensor product of localized graded modules is the tensor product of their degree-zero parts over $(S_f{)}_0$. This proves tensor compatibility of graded sheafification on degree-one-generated Proj.

The graded shift $M(-a)$ is the same underlying module with $M(-a{)}_d=M_{d-a}$. Thus $S(-a)$ has a free generator in degree $a$. Its Hilbert series is $t^a{H}_M(t)$, and multiplication by a homogeneous element of degree $a$ becomes a degree-preserving map $M(-a)\to M$.

A graded algebra over a field $k$ is a $k$-algebra $A={⨁}_{n\ge 0}{A}_n$ whose multiplication satisfies $A_i{A}_j\subseteq A_{i+j}$.

The bracket of homogeneous elements in a parity-graded associative algebra. It is a commutator if at least one element is even and an anticommutator if both are odd. This is the natural bracket for the action of an odd BRST charge on quantum operators.

A graded algebra equipped with a degree-one differential satisfying $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$ and $d^2=0$. Its cohomology inherits a graded algebra structure. In rational homotopy, a Sullivan minimal model is a free graded-commutative differential graded algebra with decomposable generator differentials.

For an even-degree generator, take a free graded $R$-module with basis $a_k$, $k\ge 0$, and multiplication $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$, with $a_0=1$ and $|a_k|=k|a|$. Associativity follows from the multinomial identity. Over $\mathbb{Z}$, $a_1^k=k!a_k$, so the algebra need not be polynomial on $a_1$.

An odd graded derivation satisfies $Q(ab)=(Qa)b+(-1{)}^{|a|}a(Qb)$ for homogeneous $a$. Its square is an even derivation; consequently checking $Q^2=0$ on algebra generators establishes nilpotence everywhere.

A graded algebra is a graded commutative algebra if $uv=(-1{)}^{pq}vu$ for homogeneous elements of a graded algebra of degrees $p,q$. In characteristic not equal to two, every odd-degree element then squares to zero. The exterior algebra and Hochschild cohomology with the Hochschild cup product are examples.

An element of a single graded component $A_n$ of a graded algebra is homogeneous of degree $n$. The degree sign rules for an exterior product or a graded Lie bracket apply first to homogeneous elements, then extend by linearity.

A Gerstenhaber algebra has an associative product making it a graded commutative algebra and a degree-minus-one graded Lie bracket. In the left convention, $[u,vw]=[u,v]w+(-1{)}^{(|u|-1)|v|}v[u,w]$. The shifted degrees $|u|-1$ determine the signs of antisymmetry and the Jacobi identity. Hochschild cohomology and the exterior algebra of polynomial derivations are examples.

On a graded algebra, a left derivation of degree $r$ satisfies the displayed rule for homogeneous $a$. A right derivation of degree $r$ satisfies $R(ab)=aR(b)+(-1{)}^{r|b|}R(a)b$. For odd $r$, these reduce to the familiar parity sign rules. The left Gerstenhaber bracket makes $[u,-]$ a left derivation of degree $|u|-1$ on Hochschild cohomology; this rule holds for the induced cohomology operations, not necessarily for their cochain representatives.

A standard graded algebra over a field $k$ has $A_0=k$ and is generated as a $k$-algebra by finitely many elements of degree one. Its Hilbert function agrees eventually with a polynomial.

For an ideal $I$, the associated graded ring is

For an increasing filtration of a module $M_n$ compatible with a filtered algebra $A$, set $\mathrm{gr}M={⨁}_n{M}_n/M_{n-1}$. Multiplication on the quotients makes it a graded module over $\mathrm{gr}A$. Its cumulative homogeneous dimensions equal $\dim M_n$ for a finite-dimensional exhaustive nonnegative filtration of a module.

For a Noetherian local ring $(A,\mathfrak{m})$, the pole dimension $d({\mathrm{gr}}_{\mathfrak{m}}(A))$ is the order of the pole at $t=1$ ofThe Hilbert-Serre theorem makes this series rational, so the order is defined.

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, $\mathrm{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.

For a Noetherian local ring $(R,\mathfrak{m})$, let $H_R(t)=\mathrm{length}(R/{\mathfrak{m}}^t)$. Its eventual Hilbert-Samuel polynomial has degree $d(R)$, equivalently the pole order at $1$ of the Hilbert series of ${\mathrm{gr}}_{\mathfrak{m}}R$. This is the local length-growth invariant, not the embedding dimension. The prime-chain lower bound for local length proves $\dim R\le d(R)$; the full local dimension theorem gives equality.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.

For an ideal of definition $I$ of a Noetherian local ring and a finitely generated module $M$, the Hilbert–Samuel function isFor sufficiently large $n$, it equals the Hilbert-Samuel polynomial.

If $M$ has Krull dimension $d$, the coefficient of $n^d$ in its Hilbert-Samuel polynomial is $e_I(M)/d!$. The positive integer $e_I(M)$ is the Hilbert–Samuel multiplicity of $M$ with respect to $I$.

For a graded module $M={⨁}_{n\ge 0}{M}_n$ whose pieces have finite length, its Hilbert series is

The Hilbert function of a graded module $M={⨁}_n{M}_n$ over a field is $n\mapsto {\dim }_k{M}_n$. Its generating function is the Hilbert series.

A quasipolynomial is a function $f:\mathbb{Z}\to \mathbb{C}$ for which there are polynomials $P_0,\dots ,P_{m-1}$ such that $f(n)=P_r(n)$ whenever $n\equiv r(\mathrm{mod}m)$. The least possible positive $m$ is its period.

Let $S=k[x_1,\dots ,x_r]$ be graded with positive degrees $d_i=\mathrm{deg}{x}_i$, and let $M$ be a finitely generated graded $S$-module with finite-dimensional graded pieces. Thenfor a Laurent polynomial $P(t)\in \mathbb{Z}[t,t^{-1}]$. For a standard grading, ${\dim }_k{M}_n$ consequently agrees with a polynomial for all sufficiently large $n$.

The proof is by induction on $r$. For $x=x_r$ of degree $d$, the exact sequence of graded modulesgivesBoth modules on the right are finitely generated over $k[x_1,\dots ,x_{r-1}]$, so induction supplies the asserted denominator.

For a homogeneous element $x$ of degree $e>0$ acting on a finite graded module, set $N=(0{:}_{M}x)$ and $C=M/xM$. The multiplication exact sequence $0\to N(-e)\to M(-e)\to M\to C\to 0$ and additivity of component lengths give $(1-t^e)P_M=P_C-t^e{P}_N$. This is the inductive step in the Hilbert-Serre theorem; the annihilator term must be kept unless multiplication by $x$ is injective.

Let a commutative $k$-algebra $R$ be generated by $x_1,\dots ,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The associated graded ring of this filtration is a standard graded algebra, andThe Hilbert-Serre theorem therefore makes ${\dim }_k{R}_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same Krull dimension.

Forwith the four displayed generators, $R_j$ has basis $Y_1^a{Y}_2^b$ with $|a|+|b|\le j$. The number of integer lattice points in this diamond isIts quadratic growth agrees with $\dim R=2$.

For a chosen additive function such as module length, the Poincare series of a graded module is the generating function of that function on its homogeneous components. In this usage it is a Hilbert series.

An ideal is an additive subgroup of a ring that absorbs multiplication by ring elements.

For ideals $I,J$ in a commutative ring, the ideal quotient is $(I:J)=\{r\in R:rJ\subseteq I\}$. Closure under subtraction and multiplication by arbitrary ring elements proves it is an ideal. If $I\subseteq J=(a)$, then $I=a(I:J)$: write $x=ar\in I$ and observe that $rJ=xR\subseteq I$. Consequently, if also $(I:J)=(b)$, then $I=(ab)$ is a principal ideal.

The whole ring $R$, viewed as an ideal. Elements $f_1,\dots ,f_r$ generate the unit ideal exactly when coefficients $g_i\in R$ give ${\sum }_i{g}_i{f}_i=1$.

If ${\sum }_i{g}_i{f}_i=1$, expand its power $r(N-1)+1$. By the pigeonhole principle, each monomial contains some $f_i^N$. Collecting terms expresses $1$ as a linear combination of these powers. This lets one use a common denominator exponent in localization arguments.

In ${\mathbb{F}}_2[u,v]/(u,v{)}^2$, the ideals $(u)$, $(v)$ and $(u+v)$ are pairwise incomparable and nonprime. Their union is the ideal $(u,v)$: the three lines cover its two-dimensional vector space over ${\mathbb{F}}_2$. This shows why a primality hypothesis matters in prime avoidance.

For an ideal $I\subseteq R$, its conormal module is $I/I^2$, naturally an $R/I$-module. It records generators and relations to first order along the quotient $R/I$. If $I$ is the maximal ideal of a local ring, it is the cotangent space of a local ring over the residue field. For the augmentation ideal $(X_1-1,\dots ,X_r-1)$ of $A[X_1^{\pm 1},\dots ,X_r^{\pm 1}]$, reduction modulo $I^2$ gives $I/I^2\cong A^r$, with basis the classes of $X_i-1$.

A proper ideal $I$ is irreducible if $I=J\cap K$ implies $I=J$ or $I=K$. In a Noetherian ring, every irreducible ideal is a primary ideal; every proper ideal is a finite intersection of irreducible ideals.

In a Noetherian ring, an irreducible ideal is a primary ideal. In its quotient, if $xy=0$ and $x\ne 0$, choose $r$ after the annihilators of powers of $y$ stabilize. Then $(x)\cap (y^r)=0$, so irreducibility forces $y^r=0$. Combined with finite decomposition into irreducible ideals, this proves the Lasker–Noether theorem.

An ideal $I$ is radical when $f^m\in I$ for some positive integer $m$ implies $f\in I$. Over an algebraically closed field, the Hilbert Nullstellensatz identifies the vanishing ideal of $V(I)$ with the radical $\sqrt{I}$.

For an ideal $I$ and an element $a$ of a commutative ring,

An ideal $I$ is nilpotent when $I^r=0$ for some positive integer $r$.

An Oka family of ideals is a family $\mathcal{G}$ such that $I+(a)\in \mathcal{G}$ and $(I:a)\in \mathcal{G}$ imply $I\in \mathcal{G}$.

Every ideal maximal outside an Oka family is prime. If $xy\in I$ while $x,y\notin I$, both $I+(x)$ and $(I:x)$ properly contain $I$, contradicting the Oka condition.

A two-sided ideal $I$ of a noncommutative ring satisfies $RI\subseteq I$ and $IR\subseteq I$.

A square-zero ideal is an ideal $I$ satisfying $I^2=0$, so every product of two elements of $I$ vanishes.

For a unital associative algebra $A$ and prescribed $A$-bimodule $M$, an extension is $0\to M\to E\to A\to 0$ where $E$ is unital, $E\to A$ is unital, $M$ is a square-zero ideal, and the induced actions are the prescribed ones. Equivalences induce the identity on $A$ and $M$. A unital linear section produces a normalized Hochschild cocycle $\mu (a,b)=s(a)s(b)-s(ab)$. Conversely, $(a,m)(b,n)=(ab,an+mb+\mu (a,b))$ on $A\oplus M$ defines the extension. Changing section changes $\mu$ by a coboundary, giving classification by $H{H}^2(A,M)$.

If $I^2=0$, then $1+I=\{1+a:a\in I\}$ is an abelian subgroup of the unit group. Multiplication satisfies $(1+a)(1+b)=1+a+b$, and $(1+a{)}^{-1}=1-a$, so $a\mapsto 1+a$ identifies the additive group of $I$ with $1+I$.

An ideal $I$ is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$.

The radical $\sqrt{I}=\{x:x^n\in I\text{ for some }n\ge 1\}$ of a primary ideal is prime. In a number ring this forces a nonzero primary ideal to have only one prime in its unique factorization.

A minimal primary decomposition of an ideal $I$ is an irredundant finite intersection $I={\bigcap }_i{Q}_i$ of primary ideals whose radicals are pairwise distinct.

The distinct radicals in a minimal primary decomposition are exactly the associated primes of a module $R/I$, and hence are invariant. The injection into the sum of primary quotients proves one inclusion; an element supported on one component, supplied by irredundancy, proves the other. This does not imply uniqueness of embedded primary components.

A component in a minimal primary decomposition is embedded if its radical of an ideal strictly contains another radical occurring in that decomposition. Such components need not be unique. For example,in $k[x,y]$ gives different $(x,y)$-primary ideals as embedded components, with the same isolated component $(x)$.

An isolated prime of a minimal primary decomposition is a minimal member of the set of radicals of its primary components.

In a Noetherian ring, the primary component belonging to each isolated prime $\mathfrak{p}$ is uniquely determined byEmbedded primary components need not be unique.

A set $S\subseteq I$ generates an ideal $I$ when every element of $I$ is a finite sum of ring multiples of elements of $S$.

A principal ideal is generated by one element: $(a)=\{ra:r\in R\}$.

The product $IJ$ consists of finite sums of products $xy$ with $x\in I$ and $y\in J$.

An ideal $I$ is idempotent when $I^2=I$. Then $I^j=I$ for every $j\ge 1$, so a nonzero idempotent ideal gives a nonzero intersection of all its powers. A semigroup algebra with arbitrarily divisible positive exponents supplies examples in a non-Noetherian integral domain. If $I$ is finitely generated, the determinant trick applied to $I=I^2$ produces $r\in I$ such that $(1+r)I=0$.

The set-theoretic intersection of any family of ideals is again an ideal.

Ideals $I$ and $J$ are comaximal when $I+J=R$.

A prime-power ideal has the form ${\mathfrak{p}}^m$ for a prime ideal $\mathfrak{p}$ and positive integer $m$.

For finitely many distinct prime ideals, one can prescribe finite valuation data simultaneously by the Chinese remainder theorem.

Reduction modulo an ideal $I$ is the quotient homomorphism $R\to R/I$.

For a two-sided ideal $I$ of a ring $R$, the quotient $R/I$ has addition and multiplication induced from $R$.

An ideal in a graded ring is homogeneous when it is generated by homogeneous elements, equivalently when every homogeneous component of each of its elements also belongs to the ideal.

The radical of an ideal $I$ isAn ideal equal to its radical is a radical ideal.

In a Noetherian ring, every radical of an ideal is an intersection of finitely many prime ideals. A maximal counterexample ideal $J$ cannot be prime. Choose $a,b\notin J$ with $ab\in J$; the strictly larger ideals $J+(a)$ and $J+(b)$ satisfy the result, and $\sqrt{J}=\sqrt{J+(a)}\cap \sqrt{J+(b)}$. This contradicts maximality. The unit ideal uses an empty intersection.

The nilradical of a commutative ring is its ideal of nilpotent elements, equivalently the radical of an ideal applied to its zero ideal.

In $R=k[t_1,t_2,\dots ]/(t_1^2,t_2^2,\dots )$, the nilradical is generated by all ${\bar{t}}_i$. Every element uses finitely many variables and is nilpotent, while each square-free product ${\bar{t}}_1\cdots {\bar{t}}_j$ is nonzero. Thus this nilradical is not a nilpotent ideal. A Noetherian ring cannot have this behavior: finitely many nilpotent generators give a nilpotent ideal.

For every commutative ring $R$,Every nilpotent element lies in every prime ideal. If $a$ is not nilpotent, an ideal maximal among those disjoint from $\{1,a,a^2,\dots \}$ is a prime ideal avoiding $a$, by the Zorn lemma and the maximality argument.

A ring homomorphism preserves addition, multiplication, and, under the convention used here, the multiplicative identity.

A ring homomorphism from a ring to itself. Surjectivity need not imply injectivity in general, but a surjective ring endomorphism of a Noetherian ring is a ring automorphism.

A bijective ring endomorphism. Its inverse is again a ring homomorphism, so it preserves the ring structure in both directions.

The kernel of a ring homomorphism $\phi :R\to S$ is the ideal $\{r\in R:\phi (r)=0\}$. The ring homomorphism is injective exactly when this ideal is zero.

A ring homomorphism transports ideals forward by extension and backward by inverse image, or contraction.