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A ring is semilocal when its quotient by the Jacobson radical is a semisimple ring that is Artinian. Equivalently, that quotient is a right Artinian ring: it has zero Jacobson radical, and the Artinian structure theorem makes it semisimple. A commutative ring is semilocal exactly when it has finitely many maximal ideals.

A ring is semiprime when it has no nonzero nilpotent ideals. Equivalently, a two-sided ideal with square zero must be zero. This is weaker than having no nonzero nilpotent elements: a matrix ring over a field is semiprime and can have nilpotent matrices.

If the left regular module of a unital ring is semisimple, then every left module is semisimple. Every module is a quotient of a direct sum of copies of the regular module, hence a quotient of a sum of simple modules. Their simple or zero images span the quotient. A maximal independent direct sum of simple submodules must span: otherwise one of these simple images has zero intersection with it and can be added. Thus the quotient is a direct sum of simples. For an infinite sum the maximality argument uses Zorn's lemma.

Adjoining an inverse of $t$ to the skew polynomial ring of an automorphism allows finite sums ${\sum }_{j\in \mathbb{Z}}{a}_j{t}^j$, with $ta=\sigma (a)t$. Every element can be moved into the polynomial subring by multiplying by a sufficiently large power of $t$. Consequently, for a right ideal $I$ of this ring, $I$ is generated by its intersection with the polynomial subring: if $f{t}^N$ lies there, then $f=(f{t}^N)t^{-N}$. This proves transfer of right Noetherianity from the polynomial subring, and multiplication on the left proves the left-handed version.

For an automorphism $\sigma$ of a unital ring $A$, this ring consists of finite sums ${\sum }_{j\ge 0}{a}_j{t}^j$, with multiplication determined by $ta=\sigma (a)t$. Both left and right coefficient normal forms are available because $\sigma$ is invertible. The noncommutative Hilbert basis theorem implies that it is a right Noetherian ring when $A$ is, and a left Noetherian ring when $A$ is. Invertibility of $\sigma$ matters in this two-sided formulation.

A square root of unity is an element $x$ of a ring with identity satisfying $x^2=1$. In a field of characteristic different from two, the only possibilities are $1$ and $-1$: the difference of squares identity $(x-1)(x+1)=0$ has no other roots because a field has no zero divisors. In a general ring more solutions may occur; modulo eight the four residue classes $1,3,5,7$ all have square one.

The sum of two ideals $I,J$ of a ring isIt is the smallest ideal containing both $I$ and $J$: it is closed under subtraction and multiplication by ring elements, and any ideal containing $I$ and $J$ contains every such sum. The ideals are comaximal ideals when $I+J=R$, equivalently when $1=a+b$ for some $a\in I$ and $b\in J$.

For a nonzero left finitely generated module over a left Noetherian ring, take a finite projective resolution with finite projective modules. Vanishing of its top Ext functor would make the dual of its final injection surjective. These duals are right finite projective modules, so the surjection splits. Dualizing back and using finite projective double dual modules splits the original injection and shortens the resolution, contradicting the definition of projective dimension. When the dimension is zero, a coordinate map of a finite free module containing the nonzero projective summand supplies a nonzero map to $R$. The left/right distinction in the duals is essential for a noncommutative ring.

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.

Given a dual pair of free lattices $N,M$ and a rational fan, glue $\mathrm{Spec}A[{\sigma }^{\vee }\cap M]$ over common-face charts. The monoid rings are semigroup algebras. A lattice map respecting cones gives a toric morphism. This construction works over a commutative ring $A$, including $\mathbb{Z}$, and also for an infinite fan in toric geometry.

// Target: toric-geometry.bigb

For a module $M$ over an integral domain $R$, its torsion submodule isThis is a submodule. If $rm=0$ and $sn=0$ with $r,s\ne 0$, then $rs(m+n)=0$ and $rs\ne 0$; also $r(am)=a(rm)=0$ for every $a\in R$. The integral domain hypothesis supplies both the nonzero product and commutativity. These are the conventions of ^[ref].

A mixed example is the direct sum $M={\mathbb{Z}}^2\oplus \mathbb{Z}/6\mathbb{Z}$. If a nonzero integer kills $(a,b,\overline{c})$, then $a=b=0$. Conversely, $6$ kills every $(0,0,\overline{c})$. ThusThe torsion submodule isolates the finite-order part while the quotient module keeps the independent integer directions. In general $M/T(M)$ is a torsion-free module: if $r(m+T(M))=0$ with $r\ne 0$, some $s\ne 0$ kills $rm$, so $sr\ne 0$ kills $m$.

The hypothesis matters. In the ring $R=\mathbb{Z}/6\mathbb{Z}$ viewed as a module over itself, the elements $\overline{2}$ and $\overline{3}$ are killed by nonzero scalars $\overline{3}$ and $\overline{2}$, but their sum $\overline{5}$ is a unit and is killed by no nonzero scalar. Thus using arbitrary nonzero scalars in a ring with zero divisors need not give a submodule. A common broader convention uses regular elements of a ring as annihilators; the integral domain convention here avoids that ambiguity.

An element $u$ of a ring is unipotent when $u-1$ is nilpotent. For a matrix, this says that every eigenvalue is one. A unipotent algebraic group consists of unipotent elements in a faithful linear representation.

A tuple in a direct product of rings is a unit exactly when each coordinate is a unit. Indeed, multiplying by an inverse tuple gives an inverse in each coordinate; conversely, the coordinate inverses form an inverse tuple. This identifies the unit group with the direct product of groups of the individual unit groups, for finite or infinite families of unital rings.

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 ${\sum }_i{g}_i{f}_i=1$.

A polynomial in one indeterminate; its coefficients lie in a specified ring. For example, $3x^2+1$ is univariate, whereas $x+y$ involves two indeterminates.

The ring with a single element, which is both its additive and multiplicative identity. If a unital ring has $0=1$, then every element $r$ equals $r1=r0=0$, so it is this ring. Under the usual definition it is excluded from integral domains and fields. The quotient ring $\mathbb{Z}/(1)$ is an example.