Lecture 2: Semisimple Modules, Socles, Artinian Rings, Wedderburn’s Theorem

Lecture 2: Semisimple Modules, Socles, Artinian Rings, Wedderburn’s Theorem 2 Semisimple modules, socles, Artinian rings, Wedderburn’s Theorem 2.1 More on semisimple modules Example 2.1: Let 𝐷 be a skew field. Then 𝐷 𝑛 is a simple module over Mat𝑛 (𝐷 ): given any nonzero vector 𝑣 ∈ 𝐷 𝑛 , there’s a change of basis matrix 𝑀 such that 𝑀𝑣 = (1, 0, . . . , 0), and we can then use permutation matrices to get all the other basis vectors. Therefore, Mat𝑛 (𝐷 ) (𝑣 ) = 𝐷 𝑛 . Corollary 2.2: Subquotients and sums of semisimple modules are semisimple. Proof. First, we show that submodules of semisimple modules are semisimple. Let 𝑀  submodule. Then by Lemma 1.33, 𝑁 ⊕ 𝑖 ∈ 𝐽 𝐿𝑖  𝑀 . Therefore, the composition 𝑁 ↩→ 𝑁 ⊕  𝑖∈𝐽 𝐿𝑖  𝑀 ↠   𝑖 ∈𝐼 𝐿𝑖 and 𝑁 ⊂ 𝑀 a 𝐿𝑖 𝑖 ∈ 𝐽 \𝐼 is an isomorphism and 𝑁 is semisimple.  Then quotients of semisimple 𝑀 are of the form 𝑀 /𝑁 for 𝑁 a submodule, so by the above 𝑀 /𝑁  𝑖 ∈ 𝐽 𝐿𝑖 and is semisimple.     𝑀𝑖 ↠ 𝑀𝑖 , so 𝑀𝑖 is a quotient of the semisimple Finally, 𝑀𝑖 is semisimple because there is a surjection □ module 𝑀𝑖 . 𝑛 Example 2.3: Mat𝑛 (𝐷 ) is semisimple over itself. It can be decomposed as 𝑖=1 Mat𝑛 (𝐷 ) (𝑒𝑖 ) where 𝑒𝑖 are the standard basis vectors: each summand is matrices that have zeroes everywhere except the 𝑖 th column. Therefore, Mat𝑛 (𝐷 ) (𝑒𝑖 )  𝐷 𝑛 ; combined with Example 2.1, Mat𝑛 (𝐷 ) is then semisimple. 2.2 Socles Definition 2.4: The socle of a module 𝑀 , denoted Soc(𝑀 ), is the sum of all semisimple (or simple) submodules of 𝑀 . Equivalently, it is the maximal semisimple submodule of 𝑀 . Example 2.5: Let 𝑀 = C[𝑡 ] as a C[𝑡 ]-module. Then Soc(𝑀 ) = 0. Submodules of 𝑀 are ideals in C[𝑡 ], and an ideal is simple iff it contains no other ideals. But if 𝐼 ≠ 0, 𝑡 𝐼 ⊊ 𝐼 , so (0) is the only simple submodule of 𝑀 . Example 2.6: Let 𝑀 = C[𝑡 ]/𝑡 𝑛 as a C[𝑡 ]-module. Then Soc(𝑀 ) = 𝑡 𝑛−1 𝑀 and is one-dimensional. The submodules of 𝑀 are all of the form 𝑡 𝑚 𝑀 , so they are simple iff 𝑚 = 𝑛 − 1; otherwise, 𝑡 (𝑡 𝑚 𝑀 ) ⊊ 𝑡 𝑚 𝑀 . Hence the only simple submodule of 𝑀 is 𝑡 𝑛−1 𝑀 . Example 2.7: Let 𝐺 be a finite 𝑝 -group and 𝑘 be a field of characteristic 𝑝 . Let 𝑀 = 𝑘 [𝐺 ] as a 𝑘 [𝐺 ]-module. Then Soc(𝑀 ) = 𝑘 . To see that, we will show that the only simple 𝐺 -module is 𝑘 . We will induct on the order of 𝐺 . Our base case is 𝐺 = Z/𝑝 Z. Let 𝑉 be a simple 𝐺 -module. Because (𝜎 − 1) 𝑝 = 0 for all 𝜎 ∈ 𝐺 , ker(𝜎 − 1) ≠ 0 ⇒ ker(𝜎 − 1) = 𝑉 . So 𝜎 = 1 and 𝑉 must be the trivial representation. Now suppose 𝐺 is an arbitrary 𝑝 -group and 𝑉 an irreducible 𝐺 -module. Then 𝐺 has a nontrivial center (can be shown by using the class equation), and the center must contain Z/𝑝 Z. In particular Z/𝑝 Z is a normal subgroup of 𝐺 , so 𝑉 Z/𝑝Z is a nonzero 𝐺 /(Z/𝑝 Z)-representation. By induction, it contains a copy of the trivial representation, and so 𝑉 has a 𝐺 -invariant vector. So 0 ≠ 𝑉 𝐺 ⊂ 𝑉 and 𝑉 must be trivial. 2.3 Isotypic components  For a semisimple module 𝑀  𝑖 𝐿𝑖 , the direct sum decomposition is not canonical; for example, vector spaces have many different bases. But we see that the multiplicity of each 𝐿𝑖 is fixed: the number of summands 𝐿𝑖 isomorphic to 𝐿 is dim𝐷 (Hom(𝐿, 𝑀 )), 𝐷 = End(𝐿) op . Moreover, the sum of such 𝐿𝑖 is well-defined because it is generated by the images of all maps 𝐿 → 𝑀 (in fact, all embeddings 𝐿 ↩→ 𝑀 ). Definition 2.8: Using the above notation, component of 𝑀 is the sum of the images of all embed  the 𝐿-isotypic dings 𝐿 ↩→ 𝑀 . Equivalently, if 𝑀  𝐿𝑖 , it is 𝐿𝑖 𝐿 𝐿𝑖 . Proposition 2.9: 𝑀 is semisimple iff any short exact sequence 0 → 𝑀1 → 𝑀 → 𝑀2 → 0 splits. Proof. If 𝑀 is semisimple, Lemma 1.33 and Corollary 2.2 imply that every short exact sequence of the above form splits. So suppose that every short exact sequence of the above form splits. Consider the short exact sequence 0 → Soc(𝑀 ) → 𝑀 → 𝑁 → 0; thus we can write 𝑀 = Soc(𝑀 ) ⊕ 𝑁 and the module 𝑁 has no simple submodules. Notice that any submodule of 𝑁 ′ ⊂ 𝑁 is a summand of 𝑁 : consider the complement of 𝑁 ′ + Soc(𝑀 ) in 𝑀 and project down to 𝑁 . Now take some 𝑎 ≠ 0, 𝑎 ∈ 𝑁 and let 𝑁 ′ := 𝑅𝑎 ⊂ 𝑁 . By Corollary 1.30, 𝑁 ′ has a simple quotient, say 𝐿, and by the same argument 𝐿 must be a summand of 𝑁 . But 𝑁 has no simple submodules, a contradiction. □ 2.4 Classification of semisimple rings Theorem 2.10: Every 𝑅-module is semisimple iff 𝑅 is semisimple over itself iff 𝑅 = are skew fields. (This is an augmented version of Theorem 1.34.) 𝑖=1 Mat𝑛𝑖 (𝐷𝑖 ) 𝑛 where the 𝐷𝑖 Proof. The first equivalence comes from the fact that every 𝑅-module is a quotient of a free module, so if 𝑅 is semisimple, so is 𝑅 𝐼 , and so are any quotients of 𝑅 𝐼 (see Corollary 2.2). If 𝑅 is a finite product of matrix rings, Example 2.3 implies that 𝑅 is semisimple over itself. To show the last implication, assume 𝑅 is semisimple over itself and write 𝑅 = 𝐿𝑖 . This sum  is finite because 𝑅 is cyclic (it is generated by 1), so if the sum were over an index set 𝑖 ∈ 𝐼 , we could write 1 = 𝑖 ∈ 𝐽 ⊂𝐼 𝑙𝑖 where |𝐽 | < ∞  𝐿 . (The same argument would work for any finitely generated module.) Anyway, write and 𝑙𝑖 ∈ 𝐿𝑖 , so 𝑅 = 𝑖 𝑖∈𝐽 𝑅 as the sum of its isotypic components, say  𝐿𝑗 𝑗 , 𝐿𝑗 ≠ 𝐿𝑗′ ⇔ 𝑗 ≠ 𝑗 ′. 𝑑 𝑗 ∈𝐽 We know that 𝑅 op  = End𝑅 (𝑅) = End𝑅   𝑑 𝐿𝑗 𝑗 𝑗 ∈𝐽 =  Mat𝑑 𝑗 (End𝑅 (𝐿 𝑗 )) 𝑗 ∈𝐽 and if we let 𝐷 𝑗 = (End𝑅 (𝐿 𝑗 )) op , we get an isomorphism  Mat𝑑 𝑗 (𝐷 𝑗 ). 𝑅 𝑗 ∈𝐽 □ Remark 2.11: It would seem natural to call rings 𝑅 semisimple over themselves semisimple. However, there is a separate notion of a simple ring, and not all simple rings are semisimple over themselves (see Example 2.13 below). 2.5 Simple rings and Wedderburn’s Theorem Definition 2.12: A ring 𝑅 is simple if 𝑅 has no 2-sided ideals except for 0 and 𝑅. Example 2.13: 𝑅 = C⟨𝑥 , 𝜕𝑥 ⟩ is simple but not semisimple. To see that 𝑅 is not semisimple, consider 𝑅/𝑅 (𝑥 𝜕𝑥 ). This module has a surjection to 𝑅/𝑅 (𝜕𝑥 ) that does not split (exercise). Definition 2.14: A ring 𝑅 is left (resp. right) Noetherian if every ascending chain of left (resp. right) ideals of 𝑅 stabilizes (called the ascending chain condition). Equivalently, every left (resp. right) ideal is finitely generated. Definition 2.15: A ring 𝑅 is left (resp. right) Artinian if every descending chain of left (resp. right) ideals of 𝑅 stabilizes (the descending chain condition). Warning : Being left Artinian/Noetherian is not equivalent to being right Artinian/Noetherian! Theorem 2.16 (Wedderburn): Let 𝑅 be a ring. TFAE: a) 𝑅 is simple and (either left or right) Artinian, b) every 𝑅-module is semisimple and 𝑅 has a unique simple module up to isomorphism, c) 𝑅  Mat𝑛 (𝐷 ) where 𝐷 is a skew field. Proof. The equivalence of 𝑏) and 𝑐 ) follows from Theorem 2.10: if 𝑅 is a finite product of matrix rings over skew fields, check that Mat𝑛 (𝐷 ) is simple over itself, and so 𝑅 has a unique simple module iff the product only contains one matrix ring. This also shows that 𝑐 ) implies 𝑎). So suppose that 𝑅 is left Artinian and simple. Then  𝑅 has a minimal left ideal (because any descending chain of left ideals will stabilize), call it 𝐿. Notice that 𝐿𝑅 = 𝑥 ∈𝑅 𝐿𝑥 is a nonzero two-sided ideal, hence all of 𝑅, and 𝑅 = 𝐿𝑅.  So 𝑅 as a left 𝑅-module is a quotient of 𝑥 ∈𝑅 𝐿, and 𝑅 is semisimple over itself. Thus 𝑎) implies 𝑏) by use Theorem 2.10. □