zbMATH — the first resource for mathematics

Unique factorisation of normal elements in polynomial rings. (English) Zbl 0662.16004
An element x of a ring S is normal if \(xS=Sx\), and N(S) denotes the set of non-zero normal elements of S. From now on let R be a prime left and right Noetherian ring in which every non-zero ideal contains a non-zero normal element, and let \(R^*\) be the ring of polynomials over R in one indeterminate. It is difficult to determine the elements of \(N(R^*)\) beyond saying that \(N(R^*)\) contains the graded normal elements of \(R^*\), i.e. those elements f of \(R^*\) such that \(fR=Rf\). This problem arises when trying to answer the following open question: If N(R) satisfies the unique factorisation property, is the same true for \(N(R^*)?\) The author shows that the answer is “Yes” if R is an integral domain or if R has an infinite central subfield. In general it is shown that if N(R) has unique factorisation, then \(N(R^*)\) has unique factorisation if and only if every element of \(N(R^*)\) is an associate of a graded normal element of \(R^*\).
Reviewer: A.W.Chatters

16N60 Prime and semiprime associative rings
16P40 Noetherian rings and modules (associative rings and algebras)
16U10 Integral domains (associative rings and algebras)
Full Text: DOI
[1] DOI: 10.1017/CBO9780511565960 · doi:10.1017/CBO9780511565960
[2] McConnell, Non-commutative Noetherian rings (1987)
[3] DOI: 10.1017/S0305004100061296 · Zbl 0541.16001 · doi:10.1017/S0305004100061296
[4] DOI: 10.1112/jlms/s2-33.1.22 · Zbl 0601.16001 · doi:10.1112/jlms/s2-33.1.22
[5] Cohn, Free rings and their relations (1985) · Zbl 0659.16001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.