The ascending chain condition for principal left or right ideals of skew generalized power series rings
The ascending chain condition for principal left or right ideals of skew generalized power series rings
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DOI：
10.1016/j.jalgebra.2009.03.040
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发表时间：
2009-08
影响因子：
0.9
通讯作者：
R. Mazurek;M. Ziembowski
中科院分区：
文献类型：
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作者：
R. Mazurek;M. Ziembowski
设R为环，S为单群，ω:S→End(R)为单群同态。证明了如果单群S是严格全序的或S是交换无扭消半全序的，则系数在R、指数在S的偏广义幂级数的环R[S，ω]是满足主左上升链条件的定域。当且仅当R是一个定义域时，R和S满足主左上的升链条件。对)理想，每个ω(s)都是内射的(p。是内射并保留R的非单位。作为直接结果，我们得到了幂级数环、劳伦级数环、斜幂级数环、斜劳伦级数环和广义幂级数环在主左（或右）理想上满足升链条件的定域的刻画。构造了主单侧理想上的升链条件不对称的偏广义幂级数域的例子。
Let R be a ring, S a monoid and ω:S→End(R) a monoid homomorphism. In this paper we prove that if the monoid S is strictly totally ordered or S is commutative torsion-free cancellative semisubtotally ordered, then the ring R[S,ω] of skew generalized power series with coefficients in R and exponents in S is a domain satisfying the ascending chain condition on principal left (resp. right) ideals if and only if R is a domain, R and S satisfy the ascending chain condition on principal left (resp. right) ideals and each ω(s) is injective (resp. is injective and preserves nonunits of R). As an immediate consequence we obtain characterizations of power series rings, Laurent series rings, skew power series rings, skew Laurent series rings and generalized power series rings that are domains satisfying the ascending chain condition on principal left (or right) ideals. We construct examples of skew generalized power series domains for which the ascending chain conditions on principal one-sided ideals are not symmetric.