In this note, we address some questions I had concerning rings when viewed as modules over themselves.
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We have:
Theorem: The left submodules of a ring R are precisely the left ideals, and the right submodules are precisely the right ideals.
This becomes evident when we write it out.
We can define what it means for a module to be left simple / right simple / two-sided simple in the obvious way. Thus, we can consider when a ring is left or right simple as a module.
Theorem: A ring is left simple as a module if and only if it has no nontrivial left ideals. A ring is right simple as a module if and only if it has no nontrivial right ideals.
This follows directly from the above.
Corollary: A ring is simple as a ring if and only if it is simple as a bimodule (meaning no nontrivial sub-bimodules.)
I’ve seen two different definitions of “semisimple ring,” one from my class / NCatlab and one from Wikipedia. Are these equivalent?
Wikipedia gives a definition of a ring being semisimple (as a ring) if it is semisimple over itself as a left (or equivalently right) module. (This equivalence is not obvious.) It defines a semisimple module to be a module that is a direct sum of simple submodules — this definition can be written for any side.
On the other hand, our class defines a semisimple ring to be one with vanishing Jacobson radical. (NCatlab adds the condition that the ring be Artinian, which personally I like more since that is needed generally to imply that the Artin-Weddernburn Theorem applies to all semisimple rings without extra conditions.) So can we show that Wikipedia’s definition is equivalent to a ring with vanishing Jacobson radical that is Artinian?
Actually, Wikipedia itself states this equivalence. So there we have it — we can view a ring as semisimple in terms of the standard ring-based definition, or as left semisimple in terms of direct sum of simple left submodules, or as right semisimple in terms of direct sum of simple right submodules — the notions are all equivalent in this case. This gives even more reason to adopt the definition of semisimple as requiring Artinian.
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EDIT (11/30/2023 — same day, a bit later): Actually, I have a bunch of questions about this.
We can define concepts like simple and semisimple much more generally beyond rings and modules. In category theory, semisimple objects are defined to be those that are direct sums of simple objects. Simple objects in turn are defined more generally in category theory as well as universal algebra — in category theory they are the objects X that have no quotient objects other than 0 or X (according to NCatLab), while in universal algebra they are the models which have no nontrivial congruences (according to Wikipedia.) (By the way, it looks like for abelian categories that the definition reduces to no subobjects other than 0 or X.) According to NCatLab, a simple ring isn’t an example of the category theoretic definition, but according to Wikipedia a simple ring is an example of the universal algebraic definition. So is there something that the category theoretic definition doesn’t capture from universal algebraic cases?
Either way, I think, against the grain of current terminology, it is best to define a semisimple ring as in the vein of universal algebra: as a ring that is a direct sum of simple subrings. (There is no expectation that an object’s simplicity or semisimplicity would be invariant when we consider it under different algebraic structures — that would be like assuming that homomorphisms are invariant when considering objects as different algebraic structures, which is clearly nonsensical.)
So under this revised terminology, how do the standard results read?
Theorem. If a ring has vanishing Jacobson radical and is Artinian, then it is semisimple.
This is a restatement (excluding some info) of the Wedderburn-Artin Theorem.
Theorem. If a ring is semiprime and left Artinian, then it is semisimple.
This was covered in my UCLA math class (Math 900.)
I highly suspect that the “left” doesn’t really matter and that the same is true for the right side too. I also suspect that if a ring is semisimple, then that would imply it must be Artinian (you can’t have an infinite descending chain when all the elements of the ring are covered by a finite direct sum, that too of simple subrings.) In fact, I’d suspect the following:
Conjecture. The following are equivalent for a ring R:
1. R is semisimple as a left module over itself
2. R is semisimple as a right module over itself
3. R is semisimple as a ring and is Artinian on one side
4. R is semisimple as a ring and is Artinian on both sides
5. R is semiprime and is Artinian on one side
6. R is semiprime and is Artinian on both sides
Proof: 1 and 2 are equivalent according to Wikipedia assertion. 5 and 6 are equivalent as follows. 6 clearly implies 5. Conversely, assume 5, so R is semiprime and Artinian on one side. By the version of Wedderburn-Artin theorem discussed in class (UCLA Math 900), R is a finite direct sum of simple subrings (aka semisimple.) Such a direct sum cannot have an infinite descending chain on either side (will be proven in a below lemma), hence must be Artinian on both sides, yielding 6. By similar logic, 3 and 4 are equivalent. As just mentioned, 5 implies 3. 1 implies 3 since …
It remains to continue this.
