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A new class of unique product monoids with applications to ring theory. (English) Zbl 1177.16030
Let $$(S,\leq)$$ be an ordered monoid. This means that $$\leq$$ is a partial order on $$S$$ which is compatible with the operation on $$S$$. The notion of an Artinian narrow unique product monoid is introduced. Namely, $$(S,\leq)$$ is such a monoid if for every two Artinian and narrow subsets $$A,B$$ of $$S$$ there exists an element in the product $$AB$$ which is uniquely presented in the form $$ab$$, where $$a\in A$$ and $$b\in B$$. Such monoids are unique product monoids in the standard sense, that originated from the zero divisor problems for group rings and semigroup rings. On the other hand, if $$\leq$$ is the trivial order on a monoid $$S$$, then $$S$$ is a unique product monoid if $$(S,\leq)$$ is an Artinian narrow unique product monoid.
Examples explaining relations between several possible definitions of unique product monoids are given. Conditions for a skew generalized power series ring $$R[\![S,\omega]\!]$$, where $$S$$ is an Artinian narrow unique product monoid and $$\omega\colon S\to\text{End}(R)$$ is a homomorphism, to be a domain or a reduced ring are also obtained. Here skew generalized power series rings are considered in the sense introduced by the second and the third author [in Commun. Algebra 36, No. 5, 1855-1868 (2008; Zbl 1159.16032)].

##### MSC:
 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 06F05 Ordered semigroups and monoids 16S36 Ordinary and skew polynomial rings and semigroup rings 20M25 Semigroup rings, multiplicative semigroups of rings
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##### References:
  Armendariz, E.P.: A note on extensions of Baer and P.P.-rings. J. Aust. Math. Soc. 18, 470–473 (1974) · Zbl 0292.16009  Birkenmeier, G.F., Park, J.P.: Triangular matrix representations of ring extensions. J. Algebra 265(2), 457–477 (2003) · Zbl 1054.16018  Groenewald, N.J.: A note on extensions of Bear and P.P.-rings. Publ. Inst. Math. (Beograd) (N.S.) 34(48), 71–72 (1983) · Zbl 0549.20051  Gustedt, J.: Well-quasi-ordering finite posets and formal languages. J. Comb. Theory Ser. B 65(1), 111–124 (1995) · Zbl 0829.68076  Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 2(3), 326–336 (1952) · Zbl 0047.03402  Hong, C.Y., Kim, N.K., Kwak, T.K.: Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra 151(3), 215–226 (2000) · Zbl 0982.16021  Hong, C.Y., Kim, N.K., Kwak, T.K.: On skew Armendariz rings. Commun. Algebra 31(1), 103–122 (2003) · Zbl 1042.16014  Kim, N.K., Lee, K.H., Lee, Y.: Power series rings satisfying a zero divisor property. Commun. Algebra 34(6), 2205–2218 (2006) · Zbl 1121.16037  Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3(4), 289–300 (1996) · Zbl 0859.16019  Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. J. Comb. Theory Ser. A 13, 297–305 (1972) · Zbl 0244.06002  Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Math., vol. 131. Springer, New York (1991) · Zbl 0728.16001  Liu, Z.: Armendariz rings relative to a monoid. Commun. Algebra 33(3), 649–661 (2005) · Zbl 1088.16021  Liu, Z.: Special properties of rings of generalized power series. Commun. Algebra 32(8), 3215–3226 (2004) · Zbl 1067.16064  Marks, G., Mazurek, R., Ziembowski, M.: Unification of some generalizations of Armendariz rings (2008, in preparation) · Zbl 1198.16025  Mazurek, R., Ziembowski, M.: On von Neumann regular rings of skew generalized power series, Commun. Algebra (2008, to appear) · Zbl 1159.16032  Okniński, J.: Semigroup Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 138. Dekker, New York (1991) · Zbl 0746.20049  Passman, D.S.: The Algebraic Structure of Group Rings. Krieger, Melbourne (1985). Reprint of the 1977 original · Zbl 0368.16003  Rege, M.B., Chhawchharia, S.: Armendariz rings. Proc. Jpn. Acad. Ser. A Math. Sci. 73(1), 14–17 (1997) · Zbl 0960.16038  Ribenboim, P.: Noetherian rings of generalized power series. J. Pure Appl. Algebra 79(3), 293–312 (1992) · Zbl 0761.13007  Ribenboim, P.: Semisimple rings and von Neumann regular rings of generalized power series. J. Algebra 198(2), 327–338 (1997) · Zbl 0890.16004  Strojnowski, A.: A note on u.p. groups. Commun. Algebra 8(3), 231–234 (1980) · Zbl 0423.20005  Yan, X.-F.: Special properties of rings of twisted generalized power series. Xibei Shifan Daxue Xuebao Ziran Kexue Ban 43(2), 20–23 (2007)
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