irreducible ring vs simple ring

Question

I'm reading a relatively old article (1956) concerning irreducible rings. I typed on google to see what an irreducible ring is but I'm not satisfied with what I found (https://en.wikipedia.org/wiki/Irreducible_ring). But I have this notion that an irreducible ring is actually just what we now call a simple ring.

Is this correct?

Answer 1

But I have this notion that an irreducible ring is actually just what we now call a simple ring.

I sure hope not. The term was bad enough for modules. Without any context, I would hope that it means "directly irreducible." Based on a quick search for usage from 1950-1960, it seems like this might be the case. For example in a book by Jacobson, originally published in 1953, contains this discussion which only makes sense in that context.

But that doesn't completely rule out the alternative. Maybe some discipline used the term differently at that time.

If you'd like to know what the author meant by the term, you're going to have to stop keeping your source a secret so that someone else can take a look at the context.

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Nathan Jacobson Structure of Rings

I don't have a copy handy, but if you're referring to "irreducible ring of endomorphisms" as in the link I gave above, it looks like we were not far off the mark.

I can't find the page where it is defined, but from page 260, the proof of Lemma 1 indicates that an irreducible ring of endomorphisms is one with a simple faithful right module. Today we would call this a right primitive ring.

Right primitive rings do not have to be simple but they are necessarily directly indecomposable (connected).

Answer 2

In Jacobson’s book, I’m pretty sure that “irreducible ring” has no meaning in isolation.

The instances I’ve found (I don’t have the book, so this is based on the parts I can see on Google books) are about “irreducible rings of endomorphisms”.

If $R$ is a “ring of endomorphisms” of an abelian group $A$ (i.e., $A$ is a faithful $R$-module), then “$R$ is an irreducible ring of endomorphisms” simply means that $A$ is an irreducible $R$-module.

So it’s not a property of a ring, it’s a property of a ring together with an action on an abelian group. The same ring could be “irreducible” and “reducible” for two different actions.