Vol. 27, No. 2, 1968

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ISSN: 0030-8730
Characterizing primes in some noncommutative rings

Harold G. Rutherford, II

Vol. 27 (1968), No. 2, 387–392

For a ring R with identity 1, a preprime is a nonempty subset T of R which is closed under the two binary operations, addition and multiplication, of R and with 1T. A prime of R is a preprime of R which is maximal with respect to set inclusion. A field K is locally finite if every member of K is a member of some finite subfield of K. For a finite dimensional vector space V over K let G = HomK(V,V ) denote the full ring of linear transformations of V over K. Let W and L be subspaces of V with W L V aud WL. Let T(L,W) = {α G∕α L W}. Then T(L,W) is a preprime of G. Let

𝒯 = {T(L,W )∕W, L are subspaces of V,W ⊂ L,W ⁄= L}

We will show that the primes of G are exactly those preprimes T(L,W) ∈𝒯 with dimKL = 1 + dimKW.

Mathematical Subject Classification
Primary: 16.10
Secondary: 10.00
Received: 21 September 1967
Published: 1 November 1968
Harold G. Rutherford, II