More on Subtractive Semirings: Simpleness, Perfectness, and Related Problems

Abstract

Among other results on subtractive semirings, we present complete descriptions of subtractive artinian ideal-simple (congruence-simple) semirings as well as subtractive semisimple semirings. Also, solving Problem 2 of [12] for semisimple semirings, we show that matrix semirings over a semisimple semiring R are subtractive iff R is a ring. Finally, confirming a conjecture of [11] for the classes of additively regular semisimple and additively regular subtractive artinian semirings, we show that perfect semirings in those classes are just perfect rings.

Key Words:

• Congruence-simple semirings
• Ideal-simple semirings
• Perfect semirings
• Semisimple semirings
• Subtractive semiring
• Tensor product of semimodules

2000 Mathematics Subject Classification:

• Primary 16Y60, 08A30, 08A99
• Secondary 18G05, 18G99

ACKNOWLEDGMENT

The authors would like to express their deep gratitude to the anonymous referee for careful reading of our article and a number of valuable suggestions.

Notes

Communicated by V. Gould.