A note on semi-principal ideal rings

الملخص EN

A ring R is said to be a Principal Ideal Ring (PIR) iff every ideal of R is principal, and is said to be a semi-Principal Ideal Ring (semi-PIR) iff every ideal of R is a direct sum of principal ideals.

Using elementary techniques, we obtain characterizations of PIR’s and semi-PIR’s of the type R = 1[T]/I where / = (f(T)) is the ideal generated by f(T) and where f(T) is a product of two distinct monic irreducible polynomials.

As a corollary, we obtain a result of Reiner characterizing semi-PIR’s of the type TL [71/(7^—1), with p prime.