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CERTAIN INVARIANT MULTIPLICATIVE SUBSET OF A SIMPLE ARTINIAN RING WITH INVOLUTION
Keywords:
simple Artinian ring, invariant multiplicative subset, trace, norm, involutionDOI:
https://doi.org/10.17654/0972555522009Abstract
Let $R$ be a unital simple Artinian ring with center $Z$ and involution $f$. By an invariant multiplicative subset of $R$, we mean a subset of $R$ written as $M$ with the following properties: (i) $1 \in M$, (ii) $M$ is closed under multiplication, (iii) $M$ is invariant under $f$, and (iv) $M$ is invariant under all inner automorphisms of $R$. Define the trace (res. norm) of a given subset $X$ of $R$ written as $tr X$ (res. $nr X$) to be the set of all elementary traces (res. norms) $f(x)+x (res. f(x)x)$ as $x$ ranges over $X$. In this paper, we investigate the case in which the considered invariant multiplicative subset $M$ has central trace (e.g., $tr M \subset Z$) but $M$ is not contained in $Z$. Substantive information about the structure of $R$ and the type of $f$ has been provided.
Received: August 14, 2021
Accepted: September 20, 2021
References
M. H. Bien, H. X. Hai and D. T. Hue, On the unit groups of rings with involution, Acta Mathematica Hungarica (to appear).
M. Chacron, Commuting involution, Comm. Algebra 44(9) (2016), 3951-3965.
I. N. Herstein, Topics in ring theory, Chicago Lectures in Mathematics, Chicago and London, The University of Chicago Press, 1969.
I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago and London, 1976.
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