On quasi-commutative rings.

*(English)*Zbl 1347.16039The authors define a ring \(R\) (associative with identity) to be quasi-commutative if \(ab\) is in the center of \(R\) for all \(a\in C_{f(x)}\) and \(b\in C_{g(x)}\) whenever \(f(x)\) and \(g(x)\) are in the center of the polynomial ring \(R[x]\). Here \(C_{h(x)}\) denotes the set of all coefficients of the polynomial \(h(x)\). A word of caution; the terminology “quasi-commutative” for rings or for ring elements has already been used many times in many other places with different meanings.

Many examples of quasi-commutative rings are given; in particular also ones that are not commutative. It is shown that this notion is compatible with many ring contructions. For example, a ring \(R\) is quasi-commutative if and only if \(R[x]\) is quasi-commutative. If \(D_n(R)\) denotes the \(n\times n\) upper triangular matrix ring with the same element on the diagonal, then it is shown that \(R\) is commutative if and only if \(D_2(R)\) is commutative which in turn is equivalent to \(D_2(R)\) being quasi-commutative. But for \(n\geq 3\), \(D_n(R)\) is never quasi-commutative. It is also shown that the radicals of the polynomial ring over a quasi-commutative ring have the same behaviour as if over a commutative ring, i.e., for a quasi-commutative ring \(R\), the following ideals coincide: the Jacobson radical of \(R[x]\), the Wedderburn radical of \(R[x]\), the upper nil radical of \(R[x]\), the prime radical of \(R[x]\) and the ring of polynomials over the nilradical of \(R\) (set of all nilpotent elements of \(R\)).

Many examples of quasi-commutative rings are given; in particular also ones that are not commutative. It is shown that this notion is compatible with many ring contructions. For example, a ring \(R\) is quasi-commutative if and only if \(R[x]\) is quasi-commutative. If \(D_n(R)\) denotes the \(n\times n\) upper triangular matrix ring with the same element on the diagonal, then it is shown that \(R\) is commutative if and only if \(D_2(R)\) is commutative which in turn is equivalent to \(D_2(R)\) being quasi-commutative. But for \(n\geq 3\), \(D_n(R)\) is never quasi-commutative. It is also shown that the radicals of the polynomial ring over a quasi-commutative ring have the same behaviour as if over a commutative ring, i.e., for a quasi-commutative ring \(R\), the following ideals coincide: the Jacobson radical of \(R[x]\), the Wedderburn radical of \(R[x]\), the upper nil radical of \(R[x]\), the prime radical of \(R[x]\) and the ring of polynomials over the nilradical of \(R\) (set of all nilpotent elements of \(R\)).

Reviewer: Stefan Veldsman (Port Elizabeth)

##### MSC:

16U80 | Generalizations of commutativity (associative rings and algebras) |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16U70 | Center, normalizer (invariant elements) (associative rings and algebras) |

16N80 | General radicals and associative rings |