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Noncommutative Ring Theory Notes NONCOMMUTATIVE RING THEORY NOTES Contents 1. Notational Conventions & Definitions 1 1.1. Notation 1 1.2. Definitions 1 2. Introduction 1 3. Rings with d.c.c. 7 3.1. Finite Dimensional Algebras 12 4. Nil and Nilpotent Ideals 14 5. Quotient Rings 17 5.1. Ideals in Q(R) 25 6. Division Algebras 30 6.1. Brauer Groups 33 7. Gelfand-Kirillov dimension 38 1. Notational Conventions & Definitions In these notes, R always denotes a ring with 1, unless otherwise stated. 1.1. Notation. subset, proper subset, ideal, right/left ideal, right mod- ule, bimodule 1.2. Definitions. simple, semisimple, prime, semiprime, primitive, semiprim- itive, artinian, noetherian, nil, nilpotent, regular, dedekind-finite, es- sential, uniform, goldie 2. Introduction Definition 2.1. A right R-module, MR, is an abelian group under addition together with a map MR × R ! MR, written (m; r) 7! mr, such that the following hold, for all m; n 2 MR, and all r; s 2 R: (1) (m + n)r = mr + nr (2) m(r + s) = mr + ms (3) m(rs) = (mr)s (4) m · 1 = m 1 2 NONCOMMUTATIVE RING THEORY NOTES If the ring R is understood, we usually drop the subscript and just write M in place of MR. A subgroup N of MR is a submodule if NR ⊆ N. Moreover, if N is a submodule of M, then we can form the factor module M=N in the obvious way. As a set, we have M=N = f m + N j m 2 M g, and the action of R is given by (m + N)r = mr + N. We also have the concept of an R-module homomorphism, which is a map 0 ' : MR ! MR such that '(m+n) = '(m)+'(n), and '(mr) = '(m)r, for all m; n 2 MR; r 2 R. The image of a module under an R-module homomorphism is again an R-module, and all the usual isomorphism theorems are still true. Example 2.2. • A vector space is a module over a field. • R is naturally a right R-module, in which case the submodules of RR are exactly the right ideals of R. Exercise 2.3. Take any ring R and prove that the two-sided ideals of Mn(R) are of the form Mn(I), where I is a two-sided ideal of R. Show that the analogous statement for one-sided ideals is not true. Corollary 2.4. Mn(k) = fn×n matrices over the field kg, is a simple ring. Definition 2.5. A module MR is finitely generated if there are el- ements m1; : : : ; mn 2 M such that, given m 2 M, there are ele- ments r1; : : : ; rn 2 R with m = m1r1 + ··· mnrn. That is, MR = m1R + ::: + mnR. Example 2.6. R = k[x; y], the polynomial ring in two commuting variables over the field k. The ideal I = xR+yR is a finitely generated R-module. (generated by x and y). Remark 2.7. (1) If MR is finitely generated, and ' is an R-module homomor- phism, then '(MR) is also finitely generated (by the images of the generators). (2) A submodule of a finitely generated module needn't be finitely generated. For example, let V be an infinite dimensional vector space over a field K. Let R = f (v; α) j v 2 V; α 2 K g with operations (v; α) + (w; β) = (v + w; α + β) (v; α)(w; β) = (βv + αw; αβ): NONCOMMUTATIVE RING THEORY NOTES 3 R is a ring with identity (0; 1), (and hence finitely generated by (0; 1) as a right R-module). However, (V; 0) is an ideal (and thus a right R-submodule) which is not finitely generated. RR 0 R (3) ⊂ M2(R) is a ring, and I = is a right ideal 0 Q 0 0 which is not finitely generated. (because RQ is not finitely gen- erated). Definition 2.8. We say a module MR is noetherian if every submodule of MR is finitely generated. Similarly, we say a ring R is right noetherian if RR is a noetherian module. Theorem 2.9. The following are equivalent: (1) MR is noetherian (2) MR satisfies a.c.c. ( = ascending chain condition) on submod- ules: given any ascending chain M1 ⊆ M2 ⊆ M3 ⊆ · · · of submodules, there is an integer r such that Mr = Mr+1 = Mr+2 = ··· : (3) Any nonempty family of submodules of MR has a maximal ele- ment (with respect to inclusion). Proof. (1) ) (2) Suppose we have a chain of submodules M1 ⊆ M2 ⊆ · · · S Note that Mi is a submodule of M, which by hypothesis is finitely generated, by say fm1; : : : mtg. Since this set is finite, there must be S P some r for which all of the mi are in Mr. Thus Mi = miR ⊆ Mr. Hence Mr = Mr+1 = Mr+2 = ··· , and the chain stabilizes. (2) ) (3) Suppose F is a nonempty family of submodules of MR. Choose Mi 2 F. If Mi is maximal, we're done. Otherwise we can find M2 2 F with M1 ⊂ M2. If M2 is maximal, we're done. Otherwise we can find M3 2 F with M2 ⊂ M3. Since we're assuming (2), this process must terminate, at which point we've arrived at a maximal element of F. (3) ) (1) Let N be any submodule of MR, and set F = fall finitely generated submodules of Ng: Let N 0 be a maximal element of F. If N 0 = N we're done. Otherwise N 0 ( N, so we can find an element n 2 N n N 0. Then N 0 + nR is a finitely generated submodule of N which properly contains N 0, 0 contradicting our choice of N . 4 NONCOMMUTATIVE RING THEORY NOTES Definition 2.10. We say a module MR is artinian if it satisfies d.c.c (= descending chain condition) on submodules. That is, given any descending chain M1 ⊇ M2 ⊇ M3 ⊇ · · · of submodules, there is an integer r such that Mr = Mr+1 = Mr+2 = ··· : Example 2.11. (1) Any finite dimensional vector space Vk is an artinian k-module. (2) ZZ is noetherian, but not artinian. For example the chain 2Z) 22Z) 23Z) ··· doesn't terminate. (3) If RR is a commutative integral domain with d.c.c., then R is a field. (proof: Choose 0 6= a 2 R and consider the chain of ideals aR ⊇ a2R ⊇ a3R ··· which must stabilize. So anR = an+1R for some n. Then an = an+1r for some r 2 R, and since R is a domain, we can cancel an to get 1 = ar, so a is a unit.) (4) (Q=Z)Z has a Z-submodule n Zp1 = f g 2 Q=Z j p g = 0 for some n 2 N g; which is artinian, but not noetherian. The Z-submodules of n Zp1 are all of the form Zpn = f g 2 Q=Z j p g = 0 g, and we have a strictly increasing chain Zp ⊂ Zp2 ⊂ Zp3 ⊂ · · · Proposition 2.12. The following are equivalent for a right R-module M: (1) MR has d.c.c. on submodules. (2) Any nonempty family of submodules of MR has a minimal ele- ment (with respect to inclusion). Proof. Mimic the proof in the noetherian case. Proposition 2.13. Any homomorphic image of a noetherian (resp. artinian) R-module is noetherian (resp. artinian). The converse of this result is a useful type of induction for noetherian (resp. artinian) modules. Proposition 2.14. Let N be an R-submodule of M. If N and M=N are noetherian (resp. artinian) then M is noetherian (resp. artinian). NONCOMMUTATIVE RING THEORY NOTES 5 Proof. We will do the proof in the noetherian case. The artinian case is similar. Let M1 ⊆ M2 ⊆ · · · be an ascending chain of submodules of M. Then M1 \ N ⊆ M2 \ N ⊆ · · · is an ascending chain of submodules of N, and (M1 + N)=N ⊆ (M2 + N)=N ⊆ · · · is an ascending chain of submodules of M=N. Both of these chains stabilize, so choose r 2 N large enough so that Mr \ N = Mr+1 \ N = ··· and (Mr + N)=N = (Mr + N)=N = ··· : Then Mr+1 = Mr+1 \ (Mr+1 + N) = Mr+1 \ (Mr + N) = Mr + (Mr+1 \ N) = Mr + (Mr \ N) = Mr where the third equality follows from the modular law. Thus the chain stabilizes, so M is noetherian. Proposition 2.15. A finite direct sum of noetherian (resp. artinian) modules is again noetherian (resp. artinian) Proof. The proof is by induction on the number of modules. Suppose M1; ··· ;Mn are noetherian R-modules. We view Mn as a submodule ∼ of M1 ⊕ ::: ⊕ Mn in the usual way. Then (M1 ⊕ ::: ⊕ Mn)=Mn = M1 ⊕ ::: ⊕ Mn−1 is noetherian by induction. Now apply the previous proposition. Of course, the previous proposition is not true for infinite direct sums. Definition 2.16. A ring R is called right noetherian (resp. artinian) if RR is a noetherian (resp. artinian) R-module. Note that any homomorphic image of a right noetherian (resp. ar- tinian) ring is right noetherian (resp. artinian). Proposition 2.17. If R is right noetherian (resp. artinian), then any finitely generated right R-module is noetherian (resp. artinian). 6 NONCOMMUTATIVE RING THEORY NOTES Proof. Again, we will do the noetherian case only, as the artinian case is similar. Let M be a finitely generated R-module, generated by m1; ··· ; mn. If n = 1, then M = mR. Set I = f r 2 R j mr = 0 g. Then I is a right ideal of R, and we have an R-module isomor- ∼ phism M = RR=I.
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