After discussing simple modules, the next best thing is to look at semisimple modules, which are just direct sums of simple modules. Here’s a summary of the results we’ll prove:
- A module is semisimple iff it is a sum of simple submodules.
- Quotients, sums, direct sums and submodules of semisimple modules are also semisimple.
Again, we fix a base ring R, possibly non-commutative. All modules are left modules.
Definition. An R-module M is said to be semisimple if it is a sum of simple submodules.
The key theorem we wish to prove is the following.
Theorem. Let M be a semisimple R-module and a submodule. Then we can find simple submodules (indexed by ) such that
The “direct sum” ⊕ means that every element m of M is uniquely writable as a sum where and only finitely many terms are non-zero.
This will be by Zorn’s lemma. Consider collections ∑ of simple submodules S of M such that:
is a direct sum. Note that at least one ∑ exists, i.e. ∑ = ∅ is valid (in which case we get ). [ For those who worry about set-theoretic validity, note that the collection of all such ∑ forms a bona fide set. ]
To apply Zorn’s, we need to prove that every chain of ∑’s has an upper bound.
Suppose is a chain: i.e. for any either or Let let us show that is a direct sum.
- If not, then for some But this is a finite sum, so the equality already holds in some (since the ‘s form a chain), which is a contradiction.
Thus, the chain has an upper bound. Zorn’s lemma tells us there is a maximal ∑. If , pick Since M is a sum of simple submodules, write
, where each Mk is simple.
Since we have for some k. But this means is a proper submodule of Mk, and must be zero (since Mk is simple). Hence is a direct sum, so we could have added the simple module Mk to the collection ∑, contradicting its maximality. Thus, and we’re done. ♦
Now we’re ready to prove all the necessary properties of semisimple modules.
Corollary 1. Every semisimple module M is a direct sum of simple modules.
Proof. Pick N = 0 in the theorem. ♦
Corollary 2. If each is a semisimple submodule of a module M, then so is
Proof. Each is a sum of simple modules; by definition so is N. ♦
Corollary 3. If N is a submodule of a semisimple M, then there is a submodule P of M such that .
Proof. Apply the theorem and let . ♦
Corollary 4. A submodule and quotient of a semisimple module M is semisimple.
Proof. Submodule follows from the definition of semisimplicity; quotient follows from the theorem, since is a direct sum of simple modules. ♦
Definition. The ring R is semisimple if it is a semisimple module over itself.
The main result we want to show is:
Theorem. Any module over a semisimple ring R is semisimple.
Let M be a module. If m is a non-zero element of M, take the homomorphism f : R → M, which takes r → rm. Then Rm is a submodule of M isomorphic to R/ker(f), which is a semisimple R-module since R is. Thus Rm is semisimple. Since M is a sum of semisimple submodules, M is also semisimple. ♦
Let us look at some ways to create semisimple rings.
Proposition. (i) If I is a (two-sided) ideal of semisimple ring R, then R/I is a semisimple ring.
(ii) If R and S are semisimple rings, so is R × S.
(i) Any left ideal of R/I corresponds to a left ideal of R containing I, which is a sum of simple submodules J. The image of J in R/I, i.e. (J + I)/I ≅ J / (J ∩ I), is thus either 0 or J. Either way, R/I is a sum of simple submodules.
(ii) Any left ideal M of R × S is of the form I × J, for left ideal I of R and J of S. [ To see why, multiply elements of M by (1, 0) and (0, 1). ] Since I and J are both sums of simple submodules, so is I × J. ♦
Finally, decomposing R gives us a complete list of simple R-modules.
Proposition. Let R be a semisimple ring; write as a direct sum of simple left ideals. Then any simple module M is isomorphic to some . In particular, there are only finitely many simple R-modules up to isomorphism.
[ Note: the which occur may repeat; in the extreme case, we can even have for a single simple module N; this just means N is the only simple R-module up to isomorphism. ]
We know that any simple module M is isomorphic to quotient R/I for a maximal left ideal I of R. For each i, consider
Since are both simple, or an isomorphism. If all , then so is which is absurd. Hence some is an isomorphism, which proves the first statement.
The second statement follows from the following lemma. ♦
Lemma. Writing the base ring as a direct sum of submodules only finitely many of the modules are non-zero.
Indeed, write 1 as a finite sum where For an not in this list, any gives:
So y = 0 since does not lie in the list of . ♦
1. Every field K is semisimple, since K itself is a simple K-module.
2. The ring Z is not semisimple. [ Why? ]
3. Every finite abelian group M is a product of cyclic groups. Thus M is semisimple as a Z-module if and only if it is a product of prime cyclic groups. Indeed, each prime cyclic group is clearly simple, hence so is their product. Conversely, if M contains a subgroup N isomorphic to Z/pr for r>1, then M has a unique simple subgroup pr-1Z/pr so it is not semisimple.
4. Is the ring R = R[x]/(x2) semisimple? What about S = R[x]/(x2 – 1) [ Answer: no to the first question, because its only simple left ideal is Rx, so 1 does not lie in the sum of all simple left ideals. Yes to the second, since it is isomorphic to R × R. ]
5. Let R be the ring of upper triangular real 2 × 2 matrices . Is this semisimple? [ Answer: no, let R act on the space of column vectors R2. Call this R-module M. The space N := R(1, 0)t is a simple submodule of M, but we cannot find a submodule P for which M = N ⊕ P. Thus we get a non-semisimple R-module. ]
6. Let R be the ring of real 2 × 2 matrices. Prove that R is semisimple. [ Hint: write R as a direct sum of two simple modules, which are spaces of column vectors. ]