On the Jacobson radical of a ternary semiring.
A ternary semiring is a non-empty set \(S\) together with a binary operation \(+\), called addition, and a ternary multiplication, denoted by juxtaposition, such that \((S,+)\) is a commutative semigroup satisfying the following for all \(a,b,c,d,e\in S\). (i) \(abc\in S\), (ii) \((abc)de = a(bcd)e=ab(cde)\), (iii) \((a+b)cd=acd+bcd\), (iv) \(a(b+c)d=abd+acd\), (v) \(ab(c+d)=abc+abd\). This paper considers those ternary semirings that have a zero-element, which is defined as an additive identity that annihilates a triple product regardless of its position in the product. A left, right or lateral ideal, \(I\), of a ternary semiring, \(S\), is a subsemigroup that absorbs triple products in which the element from \(I\) is on the right, left, or in the middle of the product. Notions familiar to the study of semirings are considered, including h-ideals, k-ideals and k-closures of ideals. The Bourne congruence allows for the definition of a quotient ternary semiring relative to every ideal. Right ternary semimodules are defined in the expected way and the notion of an additively cancellative semimodule is required for most of the results. The notions of the zeroid of a ternary semiring, as well as irreducible, faithful and primitive semimodules are explored, resulting in a theorem that an h-ideal \(P\) of a ternary semiring \(S\) is primitive if and only if \(P\) is the annihilator of some irreducible right ternary \(S\)-semimodule. The Jacobson radical of a ternary semiring \(S\) is shown to be the intersection of all its primitive h-ideals. Notions of strongly nilpotent ideals and strongly seminilpotent ideals of a ternary semiring \(S\) are explored relative to the Jacobson radical of \(S\).
scientific article; zbMATH DE number 2113339