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The Brown-McCoy radicals of a hemiring. (English) Zbl 0189.33102

Let \(C\) denote the class of all additively commutative semirings and with each \(S\) in \(C\) associate a fined mapping \(F_S\) of \(C\) into the collection of all non-empty subsets of \(S\) subject to the following condition: If \(S\) and \(\Sigma\) are in \(C\), if \(\Phi\colon a\to \bar a\) is a homomorphism of \(S\) onto \(\Sigma\), and if \(F_S(a)\) is the image, under \(\Phi\), of the subset \(F_(a)\) of \(S\), then \(\overline{F_S(a)}\subseteq F_\Sigma(\bar a)\). An element \(a\) in a semiring \(S\) is called \(F_S\)-regular provided \(a\in F_S(a)\), and a complex \(M\) of \(S\) is said to be an \(F_S\)-regular subset of \(S\) if every element of \(M\) is an \(F_S\)-regular element of \(S\).
The \(F\)-radical [\(FK\)-radical, \(FH\)-radical] of a semiring \(S\) is the set of all elements \(b\) in \(S\) for which the principal semi-ideal \((b)\) [\(k\)-ideal \((b)_k\), \(h\)-ideal \((b)_h\)] is an \(F_S\)-regular subset of \(S\). A hemiring \(S\) is said to be of type \((H)\) provided that if \(I\) is an \(h\)-ideal of \(S\), and \(\nu\) is the natural homomorphism of \(S\) onto \(S/I\), then the image, under \(nu\), of any \(h\)-ideal of \(S\) is an \(h\)-ideal of \(S/I\).
After restricting the mapping \(F_S\) and imposing an additional condition, the author considers the \(FH\) radical only. He characterizes the \(FH\)-radical of a hemiring of type \((H)\) as the intersection of a class of \(h\)-ideals. Finally he discusses a special case of the \(FH\)-radical, namely the \(H\)-radical of a hemiring of type \((H)\).
Reviewer: S. M. Yusuf

MSC:

16Y99 Generalizations
16Y60 Semirings
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