Unique factorisation of normal elements in polynomial rings. (English) Zbl 0662.16004

Jordan, D. A.

Proc. R. Soc. Edinb., Sect. A 110, No. 3-4, 241-247 (1988).

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

MSC:

16N60	Prime and semiprime associative rings
16P40	Noetherian rings and modules (associative rings and algebras)
16U10	Integral domains (associative rings and algebras)

Keywords:

normal elements; prime left and right Noetherian ring; ring of polynomials; graded normal elements; unique factorisation

References:

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