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On Strongly Prime Submodules Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 125–139 DOI: 10.18514/MMN.2018.1670 ON STRONGLY PRIME SUBMODULES ABDULRASOOL AZIZI Received 13 May, 2015 Abstract. Let R be a commutative ring with identity and M an R-module. A proper submodule N of M is called strongly prime [resp. strongly semiprime], if ..N Rx/ M /y N [resp. ..N Rx/ M /x N ] for x;y M implies that x N or y N C[resp. Wx N ]. Strongly primeC and stronglyW  semiprime submodules2 are studied,2 in this paper.2 2 2010 Mathematics Subject Classification: 13A10; 13C99; 13E05; 13F05; 13F15 Keywords: I -maximal submodule, prime submodule, strongly prime submodule, strongly semiprime submodule 1. INTRODUCTION Throughout this paper all rings are commutative with identity and all modules are unitary. Also we consider R to be a ring and M a unitary R-module. If N is a submodule of M; then .N M/ t R tM N : Recall from [12] that aW properD submodulef 2 j N ofgM is said to be strongly prime [resp. strongly semiprime], if ..N Rx/ M /y N [resp. ..N Rx/ M /x N ] for x;y M implies that x N orCy NW [resp.Âx N ]. C W  If N 2is a strongly prime2 [resp. strongly2 semiprime]2 submodule of M and .N M/ I; then we say that N is an I -strongly prime [resp. I -strongly semiprime]W submoduleD of M: A proper submodule N of M is prime if the condition ra N; r R and a M implies that either a N or rM N: In this case, if P .N2 M /;2we say that2 N is a P -prime submodule2 of M; and it is easy to see that PDis a primeW ideal of R (see, for example, [2–8, 10, 11, 13]). In [11], another notion of strongly prime submodules was introduced. It is easy to see that for modules over commutative rings with identity, the strongly prime submodules introduced in [11] coincide with the prime submodules. However, the strongly prime submodules in this paper are special cases of prime submodules (see (1)(2)). In [12, Proposition 1.1], it is shown that: (1) Every strongly prime submodule is strongly semiprime and also prime. c 2018 Miskolc University Press 126 ABDULRASOOL AZIZI (2) Every maximal submodule is strongly prime. A characterization of strongly prime submodules is given in (1), and there we prove that the converse of (1) is also correct, see also (2). The converse of (2) is studied in (3), (1), and (6). Prime submodules have been the subject of numerous publications in the past. So the relations between prime submodules and strongly prime submodules will be helpful in connecting the previous papers on the prime notion for modules to this new notion of prime. Especially there are a few papers studying the form of the elements of the intersection of prime submodules (for example, [3,7,10,13]). This is a motivation for studying the equality of the intersection of prime submodules with the intersection of strongly prime submodules in this paper (see (7) and (3)). The Generalized Principal Ideal Theorem (GPIT) states that if R is a Noetherian ring and P is a minimal prime ideal over an ideal I generated by n elements of R; then ht P n: The module version of GPIT related to prime submodules has been studied in [Ä6] and slightly in [5]. In [12, Theorem 2.3], it is proved that the module version of GPIT related to strongly prime submodules is true for every Noetherian flat module. This shows that the behavior of height of strongly prime submodules is closer to prime ideals in comparing with prime submodules, and strongly prime submodules behave better than prime submodules for studying the dimension theory in modules. In [12, Theorem 1.7], it is claimed that any intersection of strongly prime submod- ules is a strongly semiprime submodule. We show that this claim is incorrect (see (1) and (4)). The existence of strongly prime submodules are studied in (1), (4), (3), (9). The results (2) and (5) are devoted to studying the existence of strongly semiprime sub- modules. In (3) and (3) we show that strongly prime submodules exist rarely in injective modules, although (9) shows that they are found frequently in projective modules. We apply the notion of strongly prime submodules and show that if there exists a non-zero module over a primary ring that is both projective and injective, then the ring has only one prime ideal (see (6)). 2. STRONGLY PRIME AND STRONGLY SEMIPRIME SUBMODULES Let I be an ideal of R: A submodule N of M is called I -maximal, if .N M/ I and the submodule N is maximal with respect to this property, that is,W if L Dis a submodule of M containing N with .L M/ I; then L N (see [2] and [10, p. 1810]). W D D One can easily see that if N is I -maximal and I is a prime [resp. maximal] ideal, then N is a prime [resp. maximal] submodule (see [10, Lemma 3.2]). The following result is a generalization of [12, Proposition 1.1]. This result also shows that strongly prime submodules are exactly I -maximal submodules, where I is a prime ideal of R: STRONGLY PRIME 127 Proposition 1. Let N be a submodule of M: Then the following are equivalent: (1) N is a strongly prime submodule of M . (2) N is a strongly semiprime submodule and also a prime submodule of M: (3) N is a strongly semiprime submodule of M and .N M/ is a prime ideal. (4) N is .N M/-maximal and .N M/ is a prime idealW of R: W W Proof. ((a) (b)). See [12, Proposition 1.1]. ((b) (c)).H) Note that for every prime submodule N of M; the ideal .N M/ is prime.H) W ((c) (d)). To show that N is .N M/-maximal, suppose that L is a submodule of M containingH) N with .L M/ .NW M /: Let x be an arbitrary element of L: As N .N Rx/ L; we haveW .ND M/W ..N Rx/ M/ .L M/ .N M /; that is, ..NC Rx/ M/ .N M /:W ThenÂ..N CRx/ WM /x .NW M /xD N;W and since N is aC stronglyW semiprimeD W submodule, xC N: ThisW showsD thatW L ÂN; whih completes the proof. 2 D ((d) (a)). Let P .N M /: Since N is P -maximal and P is a prime ideal, by [10H), Lemma 3.2], NDis aW prime submodule; so if ..N Rx/ M /y N; for x M and y M N; then ..N Rx/ M/ .N M/ andC evidentlyW .N M/ ..N2 Rx/ M2 /: Hencen ..N Rx/C M/ W .N ÂM/ W P and since N is P -maximal,W  x .NC Rx/W N; and so xC N: W D W D 2 C D 2 Example 1. Let P be a maximal ideal of R and F a free R module with rank F > 1: (1) F contains a P -strongly prime submodule. (2) F contains a P -prime submodule that is not strongly prime. Proof. Let F i ˛R; and assume that i0 ˛: D ˚ 2 2 (1) We show that N P i ˛; i i0 R is a P -strongly prime submodule of F: D ˚ ˚ 2 ¤ Obviously PF i ˛P N; that is P .N F /; and as P is a maximal ideal of R; we have .N F/ ˚P:2 Now let L be a submodule W of F containing N with .L F/ .N F /: SinceW .LD F/ P is a maximal ideal, L is a P -prime submodule ofW F:DIf W W D N L; then let xi i ˛ L N; and so xi0 P: Obviously .1 ıi0i /xi i ˛ N ¤ f g 2 2 n … f g 2 2  L; and thus xi0 ıi0i i ˛ xi i ˛ .1 ıi0i /xi i ˛ L; and since L is a P -prime f g 2 D f g 2 f g 2 2 submodule of M; we have ei0 ıi0i i ˛ L: Also evidently for each j ˛; j i0; D f g 2 2 2 ¤ we have ej ıj i i ˛ N L: Therefore L F; which is a contradiction. This shows that NDis f a Pg -strongly2 2  prime submoduleD submodule of F: (2) By [4, Corollary 2.9(i)] if M is a flat module and PM M; then PM is a P -prime submodule of M: So the submodule PF is a P -prime¤ submodule of F and note that PF N and .N F/ P; hence PF is not a strongly prime submodule W D of F: Recall that an ideal I of R is semiprime if I pI: Strongly semiprime submod- ules of a module are characterized in the following.D 128 ABDULRASOOL AZIZI Proposition 2. A submodule N of M is strongly semiprime if and only if .N M/ is a semiprime ideal, and N is .N M/-maximal. W W Proof. ( ). To show that .N M/ is a semiprime ideal, on the contrary, let H) W r p.N M/ .N M /: Suppose that 1 < n is the smallest positive integer with n2 W n W n 1 n 1 r .N M /; and let ´ r M N: Then R´ r M rM; and so ..N R´/ M/2 ..NW rM / M /:2Hence: n   C W  C W ..N R´/ M /´ ..N rM / M /rn 1M C W  C W rn 1..N rM / M /M rn 1.N rM / N; D C W  C  and as N is a strongly semiprime submodule, ´ N; which is impossible.
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