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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:
 [1] Anderson, D.D.; Camillo, V., Armendariz rings and Gaussian rings, Comm. algebra, 26, 2265-2272, (1998) · Zbl 0915.13001 [2] Beachy, J.A.; Blair, W.D., Rings whose faithful left ideals are cofaithful, Pacific J. math., 58, 1-13, (1975) · Zbl 0309.16004 [3] Cedó, F., Zip rings and Mal’cev domains, Commun. algebra, 19, 1983-1991, (1991) · Zbl 0733.16007 [4] Faith, C., Rings with zero intersection property on annihilatorszip rings, Publ. mat., 33, 329-332, (1989) [5] Faith, C., Annihilator ideals, Associated primes and kasch – mccoy commutative rings, comm. algebra, 19, 1967-1982, (1991) [6] Hirano, Y., On annihilator ideals of a polynomial ring over a noncommutative ring, J. pure appl. algebra, 168, 45-52, (2002) · Zbl 1007.16020 [7] Kim, N.K.; Lee, Y., Armendariz rings and reduced rings, J. algebra, 223, 477-488, (2000) · Zbl 0957.16018 [8] 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 [9] Rege, M.B.; Chhawchharia, S., Armendariz rings, Proc. Japan acad. ser. A math. sci., 73, 14-17, (1997) · Zbl 0960.16038 [10] W.R. Scott, Divisors of zero in polynomial rings, Amer. Math. Monthly (1954) 336. [11] 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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