on O-simple semirings,semigroup semirings,and two kinds of division semirings

In this paper we consider O-simple semirings S, where O denotes the multiplicative zero of S, which may be in particular the additive neutral o of S at the same time. In this context we give some statements on matrix semirings and introduce contracted semigroup semirings in §3, a matter of interest of its own. We further use our results to compare the usual concept of division semirings with a new one introduced in 18], and we show that a corresponding theorem in 18] is in general only valid for division semirings in the usual meaning. Dedicated to E.S. Lyapin on his 70th birthday