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On weak Armendariz rings. (English) Zbl 1180.16018
Let \(R\) be a ring with 1 and \(R[x]\) the polynomial ring with an indeterminate \(x\) over \(R\). A ring \(R\) is called Armendariz if for \(\sum a_ix^i,\sum b_jx^j\in R[x]\), \((\sum a_ix^i)(\sum b_jx^j)=0\) implies that \(a_ib_j=0\) for all \(i,j\), and \(R\) is called weak Armendariz if \((a_0+a_1x)(b_0+b_1x)=0\) implies that \(a_ib_j=0\) for \(i,j=0,1\).
The authors show some relationships between an Armendariz (or a weak Armendariz) ring and other kinds of rings such as reduced rings, IFP rings (i.e., for \(a,b\in R\), \(ab=0\) implies \(aRb=0\)), and Abelian rings (i.e., all idempotents are central). Moreover, characterizations are given for an Armendariz or a weak Armendariz ring which is semiprime, von Neumann regular, right Ore, or semiprime Goldie.
Theorem. Let \(R\) be a von Neumann regular ring. Then the following are equivalent: (1) \(R\) is Armendariz. (2) \(R\) is weak Armendariz. (3) \(R\) is reduced. (4) \(R\) is IFP. (5) \(R\) is Abelian. (6) \(R\) is a subdirect product of division rings.
Theorem. Let \(R\) be a right Ore ring with the classical right quotient ring \(Q\). Then \(R\) is weak Armendariz if and only if so is \(Q\).
Theorem. Let \(R\) be a semiprime right Goldie ring with its classical right quotient ring \(Q\). Then the following are equivalent: (1) \(R\) is Armendariz. (2) \(R\) is weak Armendariz. (3) \(R\) is reduced. (4) \(Q\) is Armendariz. (5) \(Q\) is weak Armendariz.

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16N60 Prime and semiprime associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U20 Ore rings, multiplicative sets, Ore localization
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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