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 ∈ R which commutes with all elements of R. E.g., 0, 1 ∈ Z(R).

Z(R) := {z ∈ R | rz = zr ∀r ∈ R}

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 ∈ 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.