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Let A =A¡(BA2 be a semi-simple ring such that A¡ and A2 are isomorphic respectively to the full matrix rings (K)2 and (K)3 over a field K. If we denote by L¡ the simple right ideal of Ai (í = l, 2) and put X'=L¡®¿2©Lt, Y=L1®Li®L2, then the endomorphism rings of X and Y are isomorphic, but there is no semi-linear isomorphism of X onto Y. (2) The category in which "objects" are modules in 31 and "maps" are all .4-homomorphisms between modules in St will be denoted by the same letter 21. Here wedoi:10.2307/1993839 fatcat:gbifgo32mfcgfiaiv3fqulwmlm