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Extensions of zip rings. (English) Zbl 1071.16020
Throughout $$R$$ is an associative ring with identity. Denote by $$UTM_n(R)$$ the $$n\times n$$ upper triangular matrix ring over a ring $$R$$ and by $$aUTM_n(R)$$ ($$n\geq 2$$) the ring having as elements upper triangular matrices with entries in $$R$$ and all the entries on the main diagonal equal. A ring $$R$$ is called right zip provided that if the right annihilator $$r_R(X)$$ of a subset $$X$$ of $$R$$ is zero, then $$r_R(Y)=0$$ for a finite subset $$Y\subseteq X$$.
C. Faith raised [in Commun. Algebra 19, No. 7, 1867-1892 (1991; Zbl 0729.16015)] a number of questions on zip rings, in connection with which the authors establish a series of results. It is proved that a ring $$R$$ is a right (left) zip ring if and only if $$aUTM_n(R)$$ is a right (left) zip ring. An Armendariz ring $$R$$ is a right zip ring if and only if $$R[x]$$ is a right zip ring.
Given a monoid $$G$$ and a ring $$R$$, denote by $$R[G]$$ the monoid ring of $$G$$ over $$R$$. A monoid $$G$$ is called a u.p. monoid (unique product monoid) if given any two non-empty finite subsets $$A$$ and $$B$$ of $$G$$ there exists at least one $$c\in G$$ that has a unique representation in the form $$c=ab$$ with $$a\in A$$ and $$b\in B$$. The authors show that, for a commutative ring $$R$$ and a u.p. monoid $$G$$ that contains an infinite cyclic submonoid, $$R$$ is a zip ring if and only if $$R[G]$$ is a zip ring.

##### MSC:
 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16S36 Ordinary and skew polynomial rings and semigroup rings 16S50 Endomorphism rings; matrix rings 16D25 Ideals in associative algebras 16S34 Group rings 20M25 Semigroup rings, multiplicative semigroups of rings
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##### References:
  Anderson, D.D.; Camillo, V., Armendariz rings and Gaussian rings, Comm. algebra, 26, 2265-2272, (1998) · Zbl 0915.13001  Beachy, J.A.; Blair, W.D., Rings whose faithful left ideals are cofaithful, Pacific J. math., 58, 1-13, (1975) · Zbl 0309.16004  Cedó, F., Zip rings and Mal’cev domains, Commun. algebra, 19, 1983-1991, (1991) · Zbl 0733.16007  Faith, C., Rings with zero intersection property on annihilatorszip rings, Publ. mat., 33, 329-332, (1989)  Faith, C., Annihilator ideals, Associated primes and kasch – mccoy commutative rings, comm. algebra, 19, 1967-1982, (1991)  Hirano, Y., On annihilator ideals of a polynomial ring over a noncommutative ring, J. pure appl. algebra, 168, 45-52, (2002) · Zbl 1007.16020  Kim, N.K.; Lee, Y., Armendariz rings and reduced rings, J. algebra, 223, 477-488, (2000) · Zbl 0957.16018  Krempa, J.; Neiwieczerzal, D., Rings in which annihilators are ideal and their application to semigroup ring, Bull. acad. Pol. sci. ser. sci. math. astronom. phys., 25, 851-856, (1977) · Zbl 0345.16017  Rege, M.B.; Chhawchharia, S., Armendariz rings, Proc. Japan acad. ser. A math. sci., 73, 14-17, (1997) · Zbl 0960.16038  W.R. Scott, Divisors of zero in polynomial rings, Amer. Math. Monthly (1954) 336.  Zelmanowitz, J.M., The finite intersection property on annihilator right ideals, Proc. amer. math. soc., 57, 213-216, (1976) · Zbl 0333.16014
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