A.D. Bell: Abstract

Prime Ideals and Radicals in Semigroup-graded Rings


Allen D. Bell, Shubhangi S. Stalder, and Mark L. Teply

Publication Information

Proceedings of the Edinburgh Mathematical Society 39 (1996), pp. 1--25.
Math Reviews: MR 97a:16081

Abstract (in LaTex)

In this paper we study the ideal structure of the direct
limit and direct sum (with a special multiplication) of a
directed system of rings; this enables us to give
descriptions of the prime ideals and radicals of semigroup
rings and semigroup-graded rings.

We show that the ideals in the direct limit correspond to
certain families of ideals from the original rings, with
prime ideals corresponding to ``prime'' families.  We
then assume the indexing set is a semigroup $\Omega$
with preorder defined by $\alpha\lt\beta$ if $\beta$ is in
the ideal generated by $\alpha$, and we use the direct sum
to construct an $\Omega$\!-graded ring; this construction
generalizes the concept of a strong supplementary
semilattice sum of rings.  We show the prime ideals in this
direct sum correspond to prime ideals in the direct limits
taken over complements of prime ideals in $\Omega$ when two
conditions are satisfied; one consequence is that when
these conditions are satisfied, the prime ideals in the
semigroup ring $S[\Omega]$ correspond bijectively to pairs
$(\Phi,Q)$ with $\Phi$ a prime ideal of $\Omega$ and $Q$ a
prime ideal of $S$.  The two conditions are satisfied in
many bands and in any commutative semigroup in which the
powers of every element become stationary.  However, we
show that the above correspondence fails when $\Omega$ is
an infinite free band, by showing that $S[\Omega]$ is
prime whenever $S$ is.

When $\Omega$ satisfies the above-mentioned conditions,
or is an arbitrary band, we give a description of the
radical of the direct sum of a system in terms of the
radicals of the component rings for a class of radicals which
includes the Jacobson radical and the upper nil radical.
We do this by relating the semigroup-graded direct sum
to a direct sum indexed by the largest semilattice
quotient of $\Omega$, and also to the direct product
of the component rings.

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