Adjoining a universal inner inverse to a ring element

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• 作者：George M. Bergman ^{gbergman@math.berkeley.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 16S10 ; 16S15 ; secondary ; 16D40 ; 16D70 ; 16E50 ; 16U99
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：449
• 期：Complete
• 页码：355-399
• 全文大小：821 K

文摘

Let R be an associative unital algebra over a field k, let p be an element of R  , and let R^′=R〈q|pqp=p〉$R^{\prime }=R〈q|pqp=p〉$. We obtain normal forms for elements of R^′$R^{\prime }$, and for elements of R^′$R^{\prime }$-modules arising by extension of scalars from R  -modules. The details depend on where in the chain pR∩Rp⊆pR∪Rp⊆pR+Rp⊆R$pR\cap Rp\subseteq pR\cup Rp\subseteq pR+Rp\subseteq R$ the unit 1 of R first appears.

This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant.

We end with a normal form result for the algebra obtained by tying together a k-algebra R given with a nonzero element p   satisfying 1∉pR+Rp$1\notin pR+Rp$ and a k-algebra S given with a nonzero q   satisfying d70e8d862c5f568a8a4e6" title="Click to view the MathML source">1∉qS+Sq$1\notin qS+Sq$, via the pair of relations p=pqp$p=pqp$, q=qpq$q=qpq$.