Noncommutative Rings and Chain Conditions

A ring R is an additive abelian group equipped with a multiplication R × R →R, denoted by (a, b) ↦ab, such that, for all a, b, c ∈R, (i) a(bc) = (ab)c; (ii) a(b + c) = ab + ac and (b + c)a = ba + ca; (iii) there is 1 ∈R such that, for all a ∈R, 1a = a = a1. Here are some examples of rings that are not commutative.

If R and S are rings, then a ring homomorphism (or ring map) is a function ϕ : R →S such that, for all r,r′ ∈R, (i) ϕ(r + r′) = ϕ(r) + ϕ(r′); (ii) ϕ(rr′) = ϕ(r)ϕ(r′); (iii) ϕ(1) = 1. If ϕ : R →S is a ring homomorphism, then the kernel is defined as usual: ker ϕ = {r ∈R : ϕ(r) = 0}. The image is also defined as usual: im ϕ = {s ∈S : s = ϕ(r) for some r ∈R}. The kernel is always a two-sided ideal, for if ϕ(a) = 0 and r ∈R, then ϕ(ra) = ϕ(r)ϕ(a) = 0 = ϕ(a)ϕ(r) = ϕ(ar), so that a ∈ker ϕ implies both ra and ar lie in ker ϕ. On the other hand, im ϕ is only a subring of S. We can form the quotient ring R/I when I is a two-sided ideal, because the multiplication on the quotient abelian group R/I, given by (r + I)(s + I) = rs + I, is well-defined: If r + I = r′ + I and s + I = s′ + I, then rs + I = r′s′ + I. That is, if r −r′ ∈I and s −s′ ∈I, then rs −r′s′ ∈I. It is routine to check that the isomorphism theorems and the correspondence theorem hold for (noncommutative) rings. We now define R-modules when R is any, not necessarily commutative, ring. In contrast to the commutative case, there are now two different kinds of R-modules: left R-modules and right R-modules. We have already defined left R-modules (although we have been calling them R-modules until now).

A representation of a ring R is a ring homomorphism σ : R →EndZ(M), where M is an abelian group. Representations of rings can be translated into the language of modules.

A series (or a filtration) of a module M (over any ring) is a finite sequence of submodules M = M0, M1, M2, . . . , Mn = {0} for which M = M0 ⊇M1 ⊇M2 ⊇· · · ⊇Mn = {0}. The factor modules of this series are the modules M0/M1, M1/M2, . . ., Mn−1/Mn = Mn−1, and the length is the number of strict inclusions; equivalently, the length is the number of nonzero factor modules. A refinement of a series is a series M = M′ 0, M′ 1, . . . , M′ k = {0} having the original series as a subsequence. Two series of a module M are equivalent if there is a bijection between the sets of nonzero factor modules of each so that corresponding factor modules are isomorphic.

A left R-module M, over any ring R, has the ascending chain condition, abbreviated ACC, if every ascending chain of left submodules stops: If S1 ⊆S2 ⊆S3 ⊆· · · is a chain of submodules, then there is some t ≥1 with St = St+1 = St+2 = · · · . A left R-module M, over any ring R, has the descending chain condition, abbreviated DCC, if every descending chain of left submodules stops: If S1 ⊇S2 ⊇S3 ⊇· · · is a chain of submodules, then there is some t ≥1 with St = St+1 = St+2 = · · · .

A ring R is left ascending-chain-finite if it has the ACC (ascending chain condition) on left ideals: every ascending chain of left ideals I1 ⊆I2 ⊆I3 ⊆· · · stops; that is, there is some t ≥1 with It = It+1 = It+2 = · · · . We define right ACC rings similarly as those rings having the ACC on right ideals. If k is a field, then every finite-dimensional k-algebra A is both left and right ascending-chain-finite, for if dim(A) = n, then there are at most n strict inclusions in any ascending chain of left ideals or of right ideals. In particular, if G is a finite group, then kG is finitedimensional, and so it is left and right ascending-chain-finite.

A left ideal A in a ring R is nilpotent if there is some integer m ≥1 with Am = {0}. Recall that Am is the set of all sums of the form a1 · · · am, where a j ∈A for all j; that is, Am = { i ai1 · · · aim : ai j ∈A}. It follows that if A is nilpotent, then every a ∈A is nilpotent; that is, am = 0. On the other hand, if a ∈R is a nilpotent element, it does not follow that Ra, the left ideal generated by a, is a nilpotent ideal. For example, let R = Mat2(k), for some commutative ring k, and let a = 0 1 0 0 . Now a2 = 0 0 0 0 , but Ra contains e = = , which is idempotent: e2 = e. Therefore, em = e ̸= 0 for all m, and so (Re)m ̸= {0}.

A k-representation of a group G is a homomorphism σ : G →GL(V ), where V is a vector space over a field k. Note that if dim(V ) = n, then GL(V ) contains an isomorphic copy of Sn [if v1, . . . , vn is a basis of V and α ∈Sn, then there is a nonsingular linear transformation T : V →V with T (vi) = vα(i) for all i]; therefore, permutation representations are special cases of k-representations. Representations of groups can be translated into the language of kGmodules (compare the next proof with that of Proposition 8.8).

Every representation σ : R →EndZ(M), where M is an abelian group, equips M with the structure of a left R-module. Conversely, every left R-module M determines a representation σ : R →EndZ(M).

Every right R-module M is a left Rop-module, and every left R-module is a right Rop-module.

If a ring R is regarded as a left module over itself, then there is an isomorphism of rings EndR(R) ∼= Rop.

Any two series M = M0 ⊇M1 ⊇· · · ⊇Mn = {0} and M = N0 ⊇N1 ⊇· · · ⊇Nk = {0} of a module M have equivalent refinements.

A module M over any ring R has a composition series if and only if it has both chain conditions on submodules.

Any two composition series of a module M are equivalent. In particular, the length of a composition series, if one exists, is an invariant of M, called the length of M.

If V is a finitely generated left vector space over a division ring Δ, then any two bases of V have the same number of elements.

Let V = n i=1 Vi be a left R-module. If there is a left R-module L and, for each i, an isomorphism ϕi : Vi →L, then there is a ring isomorphism EndR(V ) ∼= Matn(EndR(L)).

Let an R-module M be a direct sum M = B1 ⊕· · · ⊕Bm in which HomR(Bi, B j) = {0} for all i ̸= j. Then there is a ring isomorphism EndR(M) ∼= EndR(B1) × · · · × EndR(Bm).

The following conditions on a ring R are equivalent. (i) R is left ascending-chain-finite. (ii) Every nonempty family of left ideals of R contains a maximal element. (iii) Every left ideal is finitely generated. Chain Conditions

Given a ring R, the following conditions are equivalent for x ∈R: (i) x ∈J(R); (ii) for every r ∈R, the element 1 −rx has a left inverse; that is, there is u ∈R with u(1 −rx) = 1; (iii) x(R/I) = {0} for every maximal left ideal I (equivalently, x M = {0} for every simple left R-module M). Chain Conditions

, so that 0 = u(1 −x)b = b, and this is a contradiction. • Chain Conditions The maximal-ideal intersection radical is obviously a left ideal, but it turns out to be a right ideal as well; that is, J(R) is a two-sided ideal. We begin by giving another source of two-sided ideals.

(i) J(R) = I = maximal left ideal ann(R/I), and so J(R) is a two-sided ideal in R. (ii) R/J(R) is a maximal-spectrum semisimple ring.

Every k-representation σ : G →GL(V ) equips V with the structure of a left kG-module; denote this module by V σ. Conversely, every left kG-module V determines a k-representation σ : G →GL(V ).

(i) The center of a ring R, denoted by Z(R), is the set of all those elements z ∈R commuting with everything: Z(R) = {z ∈R : zr = rz for all r ∈R}. If k is a commutative ring, then k ⊆Z(kG). (ii) If D is a division ring, then its center, Z(D), is a field. Moreover, if D× is the multiplicative group of the nonzero elements of D, then Z(D×) = Z(D)×; that is, the center of the multiplicative group D× consists of the nonzero elements of Z(D). ◀ Here are two “nonexamples” of subring.

Let k be a field and let R = Matn(k). For any ℓbetween 1 and n, let COL(ℓ) denote the ℓth columns; that is, COL(ℓ) = { A = [ai j] ∈Matn(k) : ai j = 0 for all j ̸= ℓ } . We claim that COL(ℓ) is a minimal left ideal in R. If we define COL∗(ℓ) = i̸=ℓ COL(i), then COL∗(ℓ) is a left ideal with R/COL∗(ℓ) ∼= COL(ℓ) as left R-modules. Since COL(ℓ) is a minimal left ideal, it is a simple left R-module, and hence COL∗(ℓ) is a maximal left ideal. Therefore, J(R) ⊆ " ℓ COL∗(ℓ) = {0}, so that R = Matn(k) is maximal-spectrum semisimple. ◀