## The McCoy condition on noncommutative rings.(English)Zbl 1231.16032

Let $$R$$ be a ring with 1 and $$R[x]$$ a polynomial ring with indeterminate $$x$$. Then $$R$$ is called right McCoy if for nonzero $$f(x)$$, $$g(x)$$ in $$R[x]$$, $$f(x)g(x)=0$$ implies $$f(x)r=0$$ for some nonzero $$r\in R$$. Left McCoy rings and McCoy rings are similarly defined. A ring is called strongly right McCoy if $$f(x)g(x)=0$$ implies $$f(x)r=0$$ for some nonzero $$r$$ in the right ideal of $$R$$ generated by the coefficients of $$g(x)$$, where $$f(x)\neq 0$$, $$g(x)\neq 0$$ in $$R[x]$$. A left strongly McCoy ring and a strongly McCoy ring are similarly defined. Strongly right McCoy rings are right McCoy, but the converse is not true. Relationships are given between strongly McCoy rings and other classes of rings such as Abelian rings (i.e., every idempotent is central) and duo rings (i.e., every one sided ideal is two sided). Moreover, relationships are given between strongly right McCoy rings and some extension rings.
Theorem 1. Let $$R$$ be a right Ore ring with its classical right quotient ring $$Q_r(R)$$. Then $$R$$ is strongly right McCoy if and only if so is $$Q_r(R)$$. Theorem 2. Let $$D_n(R)$$ be the ring of all $$n\times n$$ upper triangular matrices with a constant main diagonal, and $$V_n(R)\subset D_n(R)$$ with all constant diagonals parallel to the main diagonal. Then the following conditions are equivalent: (1) $$R$$ is strongly right (resp. left) McCoy; (2) $$D_2(R)$$ is strongly right (resp. left) McCoy; (3) $$V_n(R)$$ is strongly right (resp. left) McCoy for any $$n\geq 2$$. Theorem 3. If $$R$$ is right duo or reversible (i.e., $$ab=0$$ implies $$ba=0$$ for $$a,b\in R$$), then $$R[x]$$ is strongly right McCoy.

### MSC:

 16U80 Generalizations of commutativity (associative rings and algebras) 16S36 Ordinary and skew polynomial rings and semigroup rings 16S50 Endomorphism rings; matrix rings
Full Text:

### References:

 [1] DOI: 10.1080/00927879908826596 · Zbl 0929.16032 · doi:10.1080/00927879908826596 [2] DOI: 10.1016/j.jpaa.2007.06.010 · Zbl 1162.16021 · doi:10.1016/j.jpaa.2007.06.010 [3] DOI: 10.1112/S0024609399006116 · Zbl 1021.16019 · doi:10.1112/S0024609399006116 [4] Goodearl K. R., Von Neumann Regular Rings (1979) [5] DOI: 10.1017/CBO9780511841699 · doi:10.1017/CBO9780511841699 [6] DOI: 10.1016/j.jpaa.2005.01.009 · Zbl 1078.16030 · doi:10.1016/j.jpaa.2005.01.009 [7] DOI: 10.1006/jabr.1999.8017 · Zbl 0957.16018 · doi:10.1006/jabr.1999.8017 [8] DOI: 10.1016/S0022-4049(03)00109-9 · Zbl 1040.16021 · doi:10.1016/S0022-4049(03)00109-9 [9] DOI: 10.4153/CMB-1971-065-1 · Zbl 0217.34005 · doi:10.4153/CMB-1971-065-1 [10] DOI: 10.1017/S0004972700039526 · Zbl 1127.16027 · doi:10.1017/S0004972700039526 [11] DOI: 10.1016/S0022-4049(02)00070-1 · Zbl 1046.16015 · doi:10.1016/S0022-4049(02)00070-1 [12] DOI: 10.2307/2309082 · Zbl 0077.25903 · doi:10.2307/2309082 [13] DOI: 10.1016/j.jalgebra.2005.10.008 · Zbl 1110.16036 · doi:10.1016/j.jalgebra.2005.10.008 [14] DOI: 10.2307/2303094 · Zbl 0060.07703 · doi:10.2307/2303094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.