×

PS-modules over Ore extensions and skew generalized power series rings. (English) Zbl 1395.16027

Summary: A right \(R\)-module \(M_R\) is called a PS-module if its socle, \(\mathrm{Soc} \left(M_R\right)\), is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. Let \(R\) be an associative ring with identity, \(M_R\) a unitary right \(R\)-module, \(O = R \left[x; \alpha, \delta\right]\) Ore extension, \(M \left[x\right]_O\) a right \(O\)-module, \(\left(S, \leq\right)\) a strictly ordered additive monoid, \(\omega : S \rightarrow \mathrm{End} \left(R\right)\) a monoid homomorphism, \(A = \left[\left[R^{S, \leq}, \omega\right]\right]\) the skew generalized power series ring, and \(B_A = \left[\left[M^{S, \leq}\right]\right]_{\left[\left[R^{S, \leq}, \omega\right]\right]}\) the skew generalized power series module. Then, under some certain conditions, we prove the following: (1) If \(M_R\) is a right PS-module, then \(M \left[x\right]_O\) is a right PS-module. (2) If \(M_R\) is a right PS-module, then \(B_A\) is a right PS-module.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nicholson, W. K.; Watters, J. F., Rings with projective socle, Proceedings of the American Mathematical Society, 102, 3, 443-450 (1988) · Zbl 0657.16015 · doi:10.2307/2047200
[2] Zelmanowitz, J., Regular modules, Transactions of the American Mathematical Society, 163, 341-355 (1972) · Zbl 0227.16022 · doi:10.1090/S0002-9947-1972-0286843-3
[3] Weimin, X., Modules with projective socles, Rivista di Matematica della Università di Parma, 1, 5, 311-315 (1992) · Zbl 0806.16004
[4] Dingguo, W., Modules with flat socles and almost excellent extensions, Acta Mathematica Vietnamica, 21, 2, 295-301 (1996) · Zbl 0882.16016
[5] Liu, Z.; Li, F., PS-rings of generalized power series, Communications in Algebra, 26, 7, 2283-2291 (1998) · Zbl 0905.16021 · doi:10.1080/00927879808826276
[6] Liu, Z., PS-modules over rings of generalized power series, Northeastern Mathematical Journal, 18, 3, 254-260 (2002) · Zbl 1041.16036
[7] Ore, Ø., Theory of non-commutative polynomials, The Annals of Mathematics, 34, 3, 480-508 (1933) · Zbl 0007.15101 · doi:10.2307/1968173
[8] Lam, T. Y.; Leroy, A.; Matczuk, J., Primeness, semiprimeness and prime radical of Ore extensions, Communications in Algebra, 25, 8, 2459-2506 (1997) · Zbl 0879.16016 · doi:10.1080/00927879708826000
[9] Annin, S., Associated primes over Ore extension rings, Journal of Algebra and its Applications, 3, 2, 193-205 (2004) · Zbl 1060.16029 · doi:10.1142/S0219498804000782
[10] Hashemi, E., Extensions of BAEr and quasi-BAEr modules, Iranian Mathematical Society—Bulletin, 37, 1, 1-13 (2011) · Zbl 1237.16027
[11] Zhongkui, L., Triangular matrix representations of rings of generalized power series, Acta Mathematica Sinica (English Series), 22, 4, 989-998 (2006) · Zbl 1102.16027 · doi:10.1007/s10114-005-0555-z
[12] Ribenboim, P., Semisimple rings and von Neumann regular rings of generalized power series, Journal of Algebra, 198, 2, 327-338 (1997) · Zbl 0890.16004 · doi:10.1006/jabr.1997.7063
[13] Zhao, R.; Jiao, Y., Principal quasi-Baerness of modules of generalized power series, Taiwanese Journal of Mathematics, 15, 2, 711-722 (2011) · Zbl 1232.16035
[14] Zhongkui, L., A note on Hopfian modules, Communications in Algebra, 28, 6, 3031-3040 (2000) · Zbl 0952.16024 · doi:10.1080/00927870008827007
[15] Varadarajan, K., Generalized power series modules, Communications in Algebra, 29, 3, 1281-1294 (2001) · Zbl 0988.16035 · doi:10.1081/AGB-100001683
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.