Skew inverse power series rings over a ring with projective socle

Kamal Paykan

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 389-395
  • ISSN: 0011-4642

Abstract

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A ring R is called a right PS -ring if its socle, Soc ( R R ) , is projective. Nicholson and Watters have shown that if R is a right PS -ring, then so are the polynomial ring R [ x ] and power series ring R [ [ x ] ] . In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R [ [ x - 1 ; α , δ ] ] and the skew polynomial ring R [ x ; α , δ ] , where R is an associative ring equipped with an automorphism α and an α -derivation δ . Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.

How to cite

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Paykan, Kamal. "Skew inverse power series rings over a ring with projective socle." Czechoslovak Mathematical Journal 67.2 (2017): 389-395. <http://eudml.org/doc/288183>.

@article{Paykan2017,
abstract = {A ring $R$ is called a right $\rm PS$-ring if its socle, $\{\rm Soc\}(R_\{R\} )$, is projective. Nicholson and Watters have shown that if $R$ is a right $\rm PS$-ring, then so are the polynomial ring $R[x]$ and power series ring $R[[x]]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R[[x^\{-1\};\alpha , \delta ]]$ and the skew polynomial ring $R[x;\alpha , \delta ]$, where $R$ is an associative ring equipped with an automorphism $\alpha $ and an $\alpha $-derivation $\delta $. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.},
author = {Paykan, Kamal},
journal = {Czechoslovak Mathematical Journal},
keywords = {skew inverse power series ring; skew polynomial ring; annihilator; projective socle ring; flat socle ring},
language = {eng},
number = {2},
pages = {389-395},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Skew inverse power series rings over a ring with projective socle},
url = {http://eudml.org/doc/288183},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Paykan, Kamal
TI - Skew inverse power series rings over a ring with projective socle
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 389
EP - 395
AB - A ring $R$ is called a right $\rm PS$-ring if its socle, ${\rm Soc}(R_{R} )$, is projective. Nicholson and Watters have shown that if $R$ is a right $\rm PS$-ring, then so are the polynomial ring $R[x]$ and power series ring $R[[x]]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R[[x^{-1};\alpha , \delta ]]$ and the skew polynomial ring $R[x;\alpha , \delta ]$, where $R$ is an associative ring equipped with an automorphism $\alpha $ and an $\alpha $-derivation $\delta $. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.
LA - eng
KW - skew inverse power series ring; skew polynomial ring; annihilator; projective socle ring; flat socle ring
UR - http://eudml.org/doc/288183
ER -

References

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