# [seqfan] Re: "Simple" rings

W. Edwin Clark wclark at mail.usf.edu
Fri Dec 28 22:53:02 CET 2012

```I'm not sure there is a name for (associative) rings that are not direct
sums of other rings. Certainly they
are not called simple rings.( Simple rings are rings with no non-trivial
ideals and in the
case of finite rings are just matrix rings over finite fields.)  In the case
of associative rings such rings would be of prime power order since a
finite ring is
the product of it's p-subrings R_p where R_p is the set of elements of ring
R whose
additive order is a power of p.  But clearly some rings of prime power
order (p-rings) are
direct sums of other p-rings.

So perhaps you could do with a list of p-rings.

Some possibly  useful webpages:

http://home.wlu.edu/~dresdeng/smallrings/
http://en.wikipedia.org/wiki/Finite_ring

There is also the book *Finite Rings with Identity* (1974)  by Bernard A.
McDonald

--Edwin

On Fri, Dec 28, 2012 at 2:27 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> Surely this sequence is already in the OEIS, but I cannot find it:
> Number of rings of order n which cannot be expressed as the direct product
> of smaller rings.
> What are these rings called?
>
> Any of the various definitions of "ring" are of interest here.
>
> For a bonus, I'd like to loop through these (ideally, such rings with 1) in
> GAP or something similar. Is this possible? I know there are problems
> enumerating small rings so a library of known examples would be fine.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
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>
```