Journal of the
Korean Mathematical Society JKMS

Given a positive integer $n,$ a ring $R$ is said to be $n$-semi-Armen\-dariz if whenever $f^n = 0$ for a polynomial $f$ in one indeterminate over $R,$ then the product (possibly with repetitions) of any $n$ coefficients of $f$ is equal to zero. A~ring $R$ is said to be semi-Armendariz if $R$ is $n$-semi-Armendariz for every positive integer $n.$ Semi-Armendariz rings are a generalization of Armendariz rings. We characterize when certain important matrix rings are $n$-semi-Armendariz, generalizing some results of Jeon, Lee and Ryu from their paper (J. Korean Math. Soc. 47 (2010), 719--733), and we answer a problem left open in that paper.

Keywords: $n$-semi-Armendariz ring, semi-Armendariz ring, upper triangular matrix ring

MSC numbers: 16N40, 16S36, 16S50