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On the product of all nonzero elements of a finite ring. (English) Zbl 0713.16012
Let \(V(n,q)\) be the n-dimensional vector space over the q-element field and denote by \(M(n,q)\) the set of all linear mappings of \(V(n,q)\) into \(V(n,q)\). Also, denote by \(\bar M(n,q)\) the subset of \(M(n,q)\) consisting of all linear mappings of rank 1. The author determines the set of products of all nonzero elements in a finite ring R. In case \(R=M(n,q)\) this set of products is \(\bar M(n,q)\cup \{0\}\). In all other cases, with five exceptions, the set of products is \(\{0\}\). In the exceptional cases the set of products contains either a single nonzero element or a single nonzero element together with 0. This contrasts sharply with the analogous situation for finite groups, where it is known that the set of products of all elements is a whole coset of the derived group.
MSC:
16P10 Finite rings and finite-dimensional associative algebras
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
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[1] DOI: 10.1112/plms/s3-2.1.69 · Zbl 0047.17001 · doi:10.1112/plms/s3-2.1.69
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