Stable Rings and Side Divisors

Abstract

All rings are commutative rings with identity. $\tilde R$ denotes the set of all units in a ring $R$ together with 0 and it is clear that $R \backslash \tilde R = \emptyset$ if and only if $R$ is a field. In addition to some other results, it is shown that $R$ is not stable if and only if there exists a unimodular sequence $(y,u)$ in $R$ with $y \in R$ and $u \in R \backslash \tilde R$ such that $u$ is not a side divisor of $y$. For each $s \ge 1$, a sequence $a_1, a_2, \ldots , a_s , a_{s+1}$ of elements in a ring $R$ is said to be stable, whenever the ideal is $(a_1, a_2, \ldots , a_s , a_{s+1} ) = (a_1 + b_1 a_{s+1}, \ldots , a_s + b_s a_{s+1} )$ for some $b_1, b_2, \ldots , b_s$ in $R$. A sequence $a_1, a_2, \ldots , a_s , a_{s+1}$ of elements in $R$ is called a unimodular sequence provided that $(a_1, a_2, \ldots , a_s , a_{s+1} ) = R$. For any fixed positive integer $n$, we shall say $R$ is $n$-stable (simply, stable for $n=1$), whenever, for all $s \ge n$ any unimodular sequence, $a_1, a_2, \ldots , a_s , a_{s+1}$ in $R$ is stable. $u \in R \backslash \tilde R$ is said to be a side divisor of $y \in R$, if $u \vert y-z$ for some $z \in \tilde R$. Besides two other different proofs, we apply the above result to show that $R [ X ]$ is not stable for any commutative ring $R$. At the end, it is shown that any Artinian ring is stable.