On the Jacobson radical of semigroup rings of commutative semigroups

Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

The function field abstract prime number theorem

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties(i) ∂(l) = 0,∂(p) > 0 for p∈P;(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)