Congruence-simple semirings without nilpotent elements

Abstract

Let S be a multiplicatively idempotent congruence-simple semiring. We show that |S| = 2 if S has a multiplicatively absorbing element.

We also prove that if S is finite then either |S| = 2 or S expressionpproximexpressiontely equexpressionl to End(L) or S-op expressionpproximexpressiontely equexpressionl to End(L) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all.

Keywords

• Congruence
• Simple
• Semiring
• Multiplicatively idempotent
• Multiplicatively absorbing