Associated primes and primary right ideals of generalized triangular matrix rings

Abstract

Let ${}_S{M}_R$ be an (S, R)-bimodule of the rings R and S. We determine the associated primes of a formal triangular matrix ring $T=(\begin{array}{cc}R& 0\\ M& S\end{array}).$ Indeed, we show that $$Ass\left(T_T\right)=\left\{\left(\begin{array}{cc}Ass\left({(R\oplus M)}_R\right)& 0\\ M& S\end{array}\right)\right\}\cup \left\{\left(\begin{array}{cc}R& 0\\ M& Ass\left(l_S\right(M\left)\right)\end{array}\right)\right\}.$$We then obtain necessary and sufficient conditions for the tertiary decomposition theory to exist on a module over an arbitrary ring. Consequently, we classify all the tertiary right ideals of the formal triangular matrix rings.

Keywords:

• Associated primes
• generalized matrix rings
• primary module
• tertiary decomposition theory

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D15
• 16D40
• 16D70

Disclosure statement

No potential conflict of interest was reported by the authors.