Cleanness of skew generalized power series rings
Bull. Korean Math. Soc.
Published online July 31, 2020
kamal paykan
Islamic Azad University, Garmsar Branch
Abstract : A skew generalized power series ring $R[[S, \omega]]$ consists of
all functions from a strictly ordered monoid $S$ to a ring $R$ whose
support contains neither infinite descending chains nor infinite
antichains, with pointwise addition, and with multiplication given
by convolution twisted by an action $\omega$ of the monoid $S$ on
the ring $R$. Special cases of the skew generalized power series
ring construction are skew polynomial rings, skew Laurent polynomial
rings, skew power series rings, skew Laurent series rings, skew
monoid rings, skew group rings, skew Mal'cev-Neumann series rings,
the ``untwisted" versions of all of these, and generalized power
series rings. In this paper we obtain some necessary conditions on
$R$, $S$ and $\omega$ such that the skew generalized power series
ring $R[[S,\omega ]]$ is (uniquely) clean. As particular cases of
our general results we obtain new theorems on skew Mal'cev-Neumann
series rings, skew Laurent series rings, and generalized power
series rings.
Keywords : Skew generalized power series ring; (uniquely) clean ring; nil Jacobson radical; 2-primal ring.
MSC numbers : 16S99; 16U99; 16S36; 16E50.
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