Quasi-regular ring
A ring in which every element is quasi-regular. An element of an alternative (in particular, associative) ring
is called quasi-regular if there is an element
such that
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The element is called the quasi-inverse of
. If
is a ring with identity 1, then an element
is quasi-regular with quasi-inverse
if and only if the element
is invertible in
with inverse
. Every nilpotent element is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition:
. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [2].
References
[1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
[2] | E. Sasiada, P.M. Cohn, "An example of a simple radical ring" J. of Algebra , 5 : 3 (1967) pp. 373–377 |
Quasi-regular ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-regular_ring&oldid=31927