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Slides C: Semisimplicity Slides C: Semisimplicity 1. K-algebras and center Z(R). 2. Simple rings and modules 3. Artinian rings and modules 4. Semisimple modules 5. Radical of a module 6. Jacobson radical of a ring 7. Product of rings 8. Wedderburn Structure Theorem 9. Over an algebraically closed field Noncommutative rings and algebras We now return to noncommutative (but associative and unital) rings. We normally take right R-modules. Then an R-module homomorphism f : M ! N satisfies: f (xr) = f (x)r which is an \associativity rule". A left R-module is the same as a right Rop-module where Rop is the opposite ring of R with order of multiplication reversed: ab in Rop is ba in R. Definition. For K a field, a K-algebra is ring A which is also a K-module (vector space over K) so that addition is K-linear and multiplication is K-bilinear. K-algebra and center Z(R) Definition. The center Z(R) of a ring R is the set of all z 2 R which commutes with all elements of R. E.g., 0; 1 2 Z(R). Z(R) := fz 2 R j rz = zr 8r 2 Rg Proposition 1.1. Z(R) is a subring of R. Proposition 1.2. For K a field, R is a K-algebra , K ⊆ Z(R) Simple rings Definition. A ring is simple if it has no nontrivial 2-sided ideals. Examples. I Fields are simple rings. I A division ring D is simple: A division ring is a ring in which every nonzero element has an inverse. Fields are commutative division rings. I H, the ring of quaternions t + xi + yj + zk; t; x; y; z 2 R is a division ring which is not a field. I H is an R-algebra. It is a division algebra over R. I Matn(D) is simple. Matn(D) is the ring of n × n matrices over a division ring D. Simple modules Definition. A module S over any ring is simple if it has no nontrivial submodules (the only submodules are 0, S). Schur's Lemma. An R-module homomorphism f : S ! T between simple R-modules is either an isomorphism or 0. Corollary. For S a simple R-module, EndR (S) is a division ring. Example. For D a division ring, Dn is a simple module over the ∼ n ∼ op n simple ring Matn(D) = EndD (DD ) = EndD (D D ). Artinian rings and modules Definition. An R-module M is Artinian if it satisfies the descending chain condition (DCC) for submodules, i.e., given any descending sequence of submodules: N1 ⊇ N2 ⊇ N2 ⊇ · · · there is an m so that Nm = Nm+1 = ··· . Definition. A ring R is (right) Artinian if it satisfies the DCC for right ideals, i.e., RR is an Artinian module. Properties of Artinian modules Proposition. • Submodules and quotient modules of Artinian modules are Artinian. (Same as for Noetherian modules.) • Every nonzero submodule of an Artinian module contains a minimal submodule. (Dual of: Every proper submodule of a Noetherian module is contained in a maximal submodule.) Examples. (1) For R a K-algebra, any module M which is finite dimensional as K-vector space is Artinian. (2) Z is Noetherian but not Artinian. (3) Z[1=p]=Z is an Artinian Z-module which is not Noetherian. Semi-simple modules Definition. A f.g. module M is semi-simple if it satisfies any of the following equivalent conditions. (1) M is a direct sum of finitely many simple modules. (1') M is a direct sum of simple modules. (2) M is a sum of simple modules. (2') M is a sum of finitely many simple modules. (3) Every submodule of M is a direct summand. Definition. The length `(M) of a semi-simple module M is the number of simple direct summands. M = S1 ⊕ S2 ⊕ · · · ⊕ Sn ) `(M) = n Corollary. (since `(M) is well-defined) semi-simple ) Artinian and Noetherian. Radical of a module Definition. The radical rM of a module M is the intersection of all maximal proper submodules. Theorem 2.23. Let M be an Artinian module. (1) M=rM is semi-simple (2) M is semi-simple iff rM = 0. non-example. rZ = 0 but Z is not semisimple since Z is not Artinian. Radical of a ring Definition. The radical rR of a ring R is the intersection of its maximal right ideals. (i.e., rR = r(RR ).) Definition. A ring R is semisimple if every finitely generated right R-module is semisimple. (Equivalently, RR is semi-simple.) Corollary. If R is a right Artinian ring then R is semi-simple , rR = 0 Proof: By Thm 2.23: RR is semi-simple , rRR = 0 Product of rings Lemma 3.10. A module M over a product of rings R × S is the direct sum M = A ⊕ B where AR is an R-module and BS is an S-module. If M is f.g., so are A and B. By induction on n, a module M over R1 × · · · × Rn is a direct sum M = M1 ⊕ · · · ⊕ Mn where Mi is an Ri -module. Example. A module over Z × R × C is A ⊕ B ⊕ C where A is an additive group, B is a real vector space and C is a complex vector space. Wedderburn Stucture Theorem Lemma 3.12. If R; S are semi-simple rings then so is R × S. ∼ Lemma 3.13. EndR (ni Si ) = Matni (Di ). Wedderburn Theorem. A ring R is semi-simple if and only if it is a finite product of matrix rings over division rings: ∼ Y R = Matni (Di ) L where Di = EndR (Si ) and RR = ni Si . ∼ L Proof: If R is semisimple ) RR = ni Si ) (by Lemma 3.13) ∼ Y ∼ Y R = EndR (RR ) = EndR (ni Si ) = Matni (Di ) Q Conversely, Lemma 3.12 ) Matni (Di ) is semisimple. Over an algebraically closed field Theorem. The only finite dimensional division algebras over an algebraically closed field K = K is K itself. Corollary. A finite dimensional algebra R over C is semisimple iff it is a product of matrix rings over C: ∼ Y R = Matni (C): Part D: The group ring CG of a finite group G is semisimple. We will use this to study representations of finite groups..
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