Stable Range in Formal Power Series with any Number of Indeterminates

Abstract

Throughout this work (unless otherwise indicated), all rings are commutative rings with identity. Let $R$ be a ring, $\Lambda$ an index set with cardinality $\vert \Lambda \vert$ and let $\{X_\lambda \}_{\lambda \in \Lambda}$ be an arbitrary set of indeterminates over $R$. In this work for each fixed $i = 1,2 \text{ or } 3$ we show the ring $T_i=R[[\{X_\lambda\}_{\lambda \in \Lambda}]]_i$ of formal power series with $\vert \Lambda \vert$ indeterminates over $R$ is $n$-stable (respectively, a $B$-ring), if and only if $R$ is $n$-stable (respectively, a $B$-ring). For each $s \geq 1$, a sequence $(a_1, a_2, \dots , a_s, a_{s+1})$ of elements in $R$ is said to be stable whenever the ideal $(a_1, a_2, \cdots , a_s,a_{s+1}) = (a_1+b_1a_{s+1}, \cdots, a_s+b_sa_{s+1})$ for some $b_1, b_2, \cdots,b_s \in R$. A sequence $(a_1,a_2, \cdots, a_s, a_{s+1}),\ a_i \in R$, is said to be unimodular whenever ideal $(a_1, a_2, \cdots, a_s, a_{s+1}) = R$. For any fixed positive integer $n$ we shall say $n$ is in the stable range of $R$ (or simply, $R$ is $n$-stable), whenever for all $s \geq n$ any unimodular sequence $(a_1, a_2, \cdots,a_s, a_{s+1}),\ a_i \in R$, is stable. $R$ is said to be a $B$-ring, if for any unimodular sequence $(a_1, a_2, \cdots, a_s, a_{s+1}),\ s \geq 2,\ a_i \in R$ with $(a_1, a_2, \cdots, a_{s-1}) \not \subset$ Jacobson radical of $R$, there exists $b \in R$ such that $(a_1, a_2, \cdots, a_s+ba_{s+1}) = R$.