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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

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