Dauns , Prune modules 161

( ii ) Given beR\L, tsR with bRt^L in 1.3 (ii). It has to be shown that t e (R/Ly. But bR(R^ t)^L with ЬфЬ, Since L < Л is a prime right ideal, R4^L. Thus t e (R/L)^, and L < i? is a new prime.

( iii ) Suppose that sRt^L with s.teR and ^ ^ Z. Since sRtR^L, зфЬ and since L<Л is a prime submodule, tR^RiL. By hypothesis R:L^L and hence teL. Thus L<R is an old prime right ideal.

( iv ) <=: immediate from (iii). =>: If RR g L, then since L Ф Ä, L < Л is not a prime right ideal. Thus iîiîJL and by 1.7 (1), L==L. For any seR\L, sR(R:L)^L implies that iî:LgL.

( v ) . =>: Follows from (iii). (v) <=: By (ii).

1 . 9 . Definition . For a submodule К^M of a, module M, let cc, ß,y eM and seR be arbitrary. Then ^^ Af is called a new

( i ) strong semiprime if: clsR^s^ К => olR^s^ K. (ii) semiprime if: jß^Ä^^^K=> ßseK. (iii) w^a/: semiprime if: yiî^^i?^^gA^=> yü^^c/ST. (iv) i?er>' vt^^ö/: semiprime if: yÄ^^Ä^^ g Ä^ => y^ g Ä'.

Abbreviate : (i) п.s. semiprime; (ii) n. semiprime; (iii) n.w. semiprime; and (iv) n.v.w. semiprime. A module is any one of (i)—(iv) if (0) < M has that property.

1 . 10 . For a submodule K^M, for arbitrary a, j3, y e M and s eR, the conditions below are necessary and sufficient in order for ЛГ^ Af to be a new

( i ) strong semiprime : ccsRs g AT => (xR^ s^K. (ii) semiprime : ßsRs ^K=> ßseK, (iii) weak semiprime : yRsRs g A: => yR^ s g К*. (iv) very weak semiprime : yRsRs ^K=>ys€K.

Proof . That 1. 10 => 1. 9 for (i)—(iv) is clear. 1. 9 => 1. 10. (i) Given asRs^K. If also ocsR^s^K, there is nothing to prove. So assume that asR^s^K, and hence that (XSQS Ф К for some qeR^. But then asQsR^SQS^asRs^K. By 1. 9 (i), (xR^sqs^K, which contradicts that asgs ф К. Hence asR^s^K. Thus by 1. 9 (i), (xR^s^K.

( ii ) Given ßsRs g К, ßsR^s^ К, and ßsQs ф К for g eRK Then ßsgsRhgs g ßsRs g К, By 1. 9 (i), ßsQs EK,2i contradiction. Thus ßsR^s^ К, and consequently ßsE К by 1,9 (ii). The proofs of (iii) and (iv) are entirely similar and are omitted.

Frequently for modules 1. 10 and for right ideals 1. 11 (3) (ii) will be used.

1 . 11 . Remarks. (1) In 1. 9, (i) strong => (ii) semiprime => (iii) weak => (iv) very weak. Furthermore, prime => strong semiprime.

( 2 ) Trivial semiprimes: (i) M^M. (ii) A:<M if MR^K. (iii) Any K<M when Ä = {0}.