On semilocal $\textrm {OP}$-rings

Abstract:

The notion of OP-rings was introduced by D. Lissner. A commutative ring R is called an OP-ring if, for any $n \geqq 2$, any vector of ${R^n}$ is an outer product of $n - 1$ vectors of ${R^n}$. Recently J. Towber proved that any local ring is an OP-ring if and only if the maximal ideal is generated by two elements. The main result in the present paper is a generalization to semilocal rings of the above theorem proved by Towber for local rings. The author’s argument does not rely on Towber’s theorem however, and so provides a new and very elementary proof of that result.