On weak Armendariz rings.

*(English)*Zbl 1180.16018Let \(R\) 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.

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.

Reviewer: George Szeto (Peoria)

##### 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 |