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RINGS IN WHICH EVERY ELEMENT IS THE SUM OF TWO IDEMPOTENTSYASUYUKI HIRANO AND HISAO TOMINAGA Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x 3 = x. (2) If R is a Pi-ring in which every element is the sum of two idempotents, then RjP satisfies the identity x -z. (3) Let it be a semi-perfect ring in which every element is the sum of two idempotents. If is quasi-projective, then R is a finite direct sum of copies of GF[2) and/or GF(3).

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Abstract. Anderson-Smith [1] studied weakly prime ideals for a commutative ring with identity. Blair-Tsutsui [2] studied the structure of a ring in which every ideal is prime. In this paper we investigate the structure of rings, not necessarily commutative, in which all ideals are weakly prime.

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