Trlifaj, Jan Constructions and homological properties of simple von Neumann regular rings. (English) Zbl 0741.16002 Czech. Math. J. 40(115), No. 4, 583-597 (1990). A ring \(R\) is called a \(\otimes\)-ring if \(M\otimes_ RN\neq 0\) for any nonzero modules \(M_ R\) and \(_ RN\). It has been shown by P. Jambor and the author that if \(R\) is commutative (von Neumann) regular or countable regular then \(R\) is simple Artinian if and only if \(R\) is a \(\otimes\)- ring. In the present paper the author investigates the question whether or not a regular ring \(R\) is simple Artinian if and only if it is a \(\otimes\)-ring and obtains interesting results for several special cases. For example, the answer is positive if \(R\) is countable regular or of cardinality equal to \(\aleph_ 1\) and \(Ann(E)\neq 0\) for each countable infinite set \(E\) of orthogonal idempotents of \(R\). In the last part of the paper three methods of constructing non-completely reducible regular simple rings are presented. Reviewer: Dinh van Huynh (Hanoi) MSC: 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 16K40 Infinite-dimensional and general division rings Keywords:\(\otimes\)-ring; countable regular; simple Artinian; orthogonal idempotents; regular simple rings × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] F. W. Anderson, K. R. Fuller: Rings and categories of modules. Springer-Verlag, New York 1974. · Zbl 0301.16001 [2] P. C. Eklof, S. Shelah: A calculation of injective dimension over valuation domains. preprint. · Zbl 0656.13012 [3] K. R. Goodearl: Von Neumann regular rings. Pitman, London 1979. · Zbl 0411.16007 [4] P. Jambor: Hereditary tensor-orthogonal theories. Comment. Math. Univ. Caroiinae 16 (1975), 139-145. · Zbl 0303.16011 [5] J. Trlifaj: On countable von Neumann regular rings. to appear in Czech. Math. J. 41 (1991). · Zbl 0790.16011 [6] A. E. Zalesskij, A. V. Michaljov: Group rings. Contemp. problems of math. 2 (1973), 5-118 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.