On Small Primary Modules

Let be a commutative ring with an identity and be a unitary -module. We say that a non-zero submodule of is primary if for each with en either or √ and an -module is a small primary if √ = √ for each proper submodule small in . We provided and demonstrated some of the characterizations and features of these types of submodules (modules).


Introduction
A non-zero submodule of is called primary if whenever and with implies that √ or . Also, is called primary if √ = √ for each proper submodule of [1]. These two concepts were generalized by many researchers [2,3,4]. As for this research, we present and study a generalization of the concepts of small primary submodule and small primary module as follows; We call a submodule of as a small primary submodule if whenever is small in and , then either or √ , and is a small primary module if √ = √ for each proper submodule small in , where "a submodule of is called small (notationally, if for all submodules of implies [5]. This research consists of two parts; in the first part, we present the definition of small primary submodules and discuss some of their relationships with some types of the previously studied submodules and gave the conditions of equivalence between them. We also gave and demonstrated some of the characteristics and features of this type of submodules. In the second part, we present a definition of small primary modules and study and demonstrate some of their properties in detail.

ISSN: 0067-2904
2-Small Primary Submodules Definition (2.1): i) A non-zero submodule of -module is called small primary iff whenever , and such that then either or √ . ii) A proper ideal of is small primary if is a small primary submodule of -module . Remark (2.2) 1-Every primary submodule is small primary . But the converse is not true; for example: Let be a -module, then each non-zero submodule of is small primary. Since if with and . But is the only small submodule in , so . Hence However, if we take , then it is clear that is not primary. 2-Suppose that is an -module and let be an ideal of with Then is small primary -submodule of iff is a small primary submodule of . Proof: Let ̅ with ( and ̅ . But ̅ Therefore , we achieve the result. 3-Let be a hollow -module, then every small primary submodule P of a module X is primary submodule, where " An -module is called a hollow module if every non-zero submodule of is small in " [6]. Proof: Suppose that , where . But is hollow, so (m) . Since is a small primary in , hence either √ or . Therefore, is a primary submodule in . 4-If is a small prime submodule of an -module then is a small primary submodule in where "A proper submodule of an -module is called small prime iff whenever x with such that implies either or [7]. Proof: Let , with . Hence either or . But √ . So either or √ . Hence is small primary. But the converse is not true; for example: Let be a -module, then ( ̅ ) is small primary since it is primary by [1] . But ( ̅ ) is not small prime, by [7] .

5-If
is a semiprime ideal of R, then P is a small primary submodule iff it is a small prime. Proof: Since is a semiprime, so √ . Hence the result follows easily.

6-If
and P is a small primary of X , then needs not to be small primary, as the following example shows: Consider that as a -module , ̅ is small primary since is small prime by [7]. be a non-zero submodule of an R-module X.Then, the followings are equivalent: 1. A submodule P is small primary 2. (P A) is a small primary submodule of X , , A R such that AX P. 3. (P: X (a)) is a small primary submodule of X, a R such that aX P.
Proof: (1) (2): Let ax (P A) and (x) P; that is a(x) (P A), then aAx P. Since (x) X, then (Ax) X. But P is small primary, so (Ax) P or a √ by theorem (2.3). But (Ax) P implies that AX P, which is a contradiction. So a √ and hence a n X P for some . But P (P A) and hence a n X (P A). It follows that a n ϵ [ (P A): X] . Hence (P A) is a small primary. (1): By taking a = 1, so it follows easily.

Proposition(2.5): Let
 be an R-epimorphism. If P is small primary submodule of a module Y, then -1 (P) is small primary submodule of X. Proof: To prove that -1 (P) is a non-zero submodule of X, suppose that -1 (P) = X, then (X) P, which is a contradiction to the assumption. Let a R, m X such that (m) X and am -1 (P). Hence . But (m) X, so (m) Y by [8], and as P is a small pimary of , then either (m) P or a n Y P for some n Z + . If (m) P, then m -1 (P) . If a n Y P, then a n (X) P since (X) = Y. This implies that a n X -1 (P) for some n Z + . Therefore -1 (P) is small primary. Proposition (2.6): Suppose that X is an R-module, S is a multiplicative subset of R, and P is a small primary of X. Then Ps is a small primary submodule of X S . Proof: Suppose that R S and x / t X S with ax / st P S such that ( x/t) ˂˂ X S . So u S such that uax P. But (x/t) X S , so (x) X by [7]. So (ux) ˂˂ X. Since P is small primary of X, then either ux P or (a) n [P:X] for some n Z + .Therefore either ux /ut = x/t P S or (a / s) n [P:X] S [P S : X S ] for some n Z + . Therefore, P S is a small primary submodule of X S . Remark (2.7 ): If W is a small primary submodule of X, then [W : X] is not a primary ideal of R. For example: as a -module , ̅ is small primary. But 6Z = [ W:X] is not primary ideal of Z. Proposition (2.8): Let P be a non-zero submodule of R-module X. If P is a small primary submodule of X, then [P:X] is a small primary ideal of R. Proof: Suppose that uv [P:X] where u, v R such that (v) ˂˂ R. Suppose that v [P:X]. Now for any x X, define x : RX by (x) = ax. So it is clear that this function is well-defined and is a homomorphism. Since (v) ˂˂ R, so for any x X we get (vx) ˂˂ X ……. (1). But v [P:X], so m X such that vm P. But uvm P. Also by (1), (vm) ˂˂ X. Since P is a small primary submodule of X, so either vm N or u n [P:X] for some n Z + . If u n [P:X], then we are done. If vm P then this contradicts our assumption. Remark (2.9 ): If [P:X] is a small primary ideal of R, so it is not necessary that P is a small primary submodule of X. For example: as a -module , ̅̅̅̅ is not small primary see (2.2,6). But [P:X] = 12Z which is small primary ideal of Z.
Recall that an R-module X is called a mulitplication if for each submodule P of X there is an ideal A of R such that P =AX [9]. Proposition (2.10): Let be a non-zero submodule of a faithful finietly generated mulitplication Rmodule . Then is a small primary submodule of X if [P:X] is a small primary ideal of R. Proof:. Let ax P where a R , y X such that (y) X. But X is a finietly generated faithful mulitplication module, so (y) =AX and A R [8] ] for some . Therefore is a small primary of . By a similar proof , is a small primary of .

3-Small Primary Modules Definition (3.1) :i) An -module is called small primary iff √
= √ . ii) A ring is a small primary ring iff √ , . Remark (3.2) 1-If X is a primary R-module , then X is small primary . But the converse is not true; for example : as a Z -module is small primary but not primary. 2-Let X be a hollow small primary R-module, then X is primary. 3-Every small prime R-module is small primary, but the converse is not true in general; for example : as a Z -module is small primary but not small prime, by [7].

Theorem (3.3):
Suppose that X is a module, then X is small primary iff √ = √ , and Proof: It is clear.

Corollary (3.5):
A non-zero submodule of a module is a small primary submodule iff is a small primary -module. Corollary (3.6): Suppose that X is a module . Then the followings are equivalent: a-A module X is small primary. b-√ = √ and c-(0) is small primary.

Proposition (3.7):
If is a small primary -module, then is a primary ideal of Proof: Let such that and Suppose that so for some , and since implies that But is a submodule of P and implies that ( [8]. On the other hand , is small primary , so (0) is a small primary of . Then u √ . But √ = √ , hence u √ . Thus, is a primary ideal in . Proposition (3.8): If X is a small primary R-module, then a non-zroe submodule is a small primary Rmodule. Proof: Supose that is a submodule of . Suppose that So [8].
and therefore is small primary.
The following example shows that the converse is not true : Let be a -module, then is a small primary Z-module. While as a -module is not a small primary Z-module. Since ( ̅ but √ √ √ ̅̅̅

Proposition (3.9):
If is a direct summand small primary of an -module Y and √ = √ , then Y is a small primary R-module, where is the Jacobson radical of Y Proof: Supose that and (x) Then so [8]. Therefore √ = √ . But √ √ so √ = √ and therefore is small primary.

Theorem (3.10):
Suppose that is an R-module and Then X is a small primary R-module iff and are small primary R-modules.

Proof:
Let Since so where and are submodules of and , respectively [10]. But , so and [8]. Now, √ √ √ √ (since and are small primary).

Theorem (3.11): Suppose that
. Then is small primary if and only if is small primary. Proof: Let be small primary. Since , so there exists that is an R-isomorphism. Assume that . Hence and [8]. So √ = √ . But implies that √ = √ , [11]. Thus √ = √ . But it is easily that √ = √ , which completes the proof.

Proposition (3.12):
If is an R-homomorphism and is small primary such that √ √ , then is small primary. Proof: Let such that √ and Then for some so implies that √ But so [8]. Since is small primary, hence √ . But √ √ so √ and hence √ √ Therefore √ √ Thus is small primary. Corollary (3.13): Suppose that is a submodule of -module and √ √ If is small primary, then is small primary. Corollary (3.14 ): If is a small primary submodule of -module and √ √ then is small primary.
Recall that an R-module M is called coprime if for every proper submodule P of X [12] . Corollary (3.15 ): If X is a coprime -module, is a submodule of is small primary, then is small primary. Proposition (3.16 ): Let be a submodule of an -module If is small primary, so √ √ W X and . Proof: Let and Hence [8]. But is small primary, so √ √ . Therefore √ √

Corollary (3.17 ):
If is a small primary submodule of an -module , then √ √ W X and Proposition (3.18 ): If is a small submodule of an -module and √ √ W X and is small primary. Proof: Let be two submodules of X and << Then [8]. Therefore √ √ , so √ √ . Hence is small primary. Corollary (3. 19): Supose that is a small submodule of R-module Then √ √ W X and is a small primary in X. Corollary (3.20 ): Supose that is a small submodule of an -module Then √ √ W X and is a small primary in X. Corollary (3. 21): Supose that is a submodule of a hollow -module Then √ √ W X and is a small primary in X. Theorem (3.22): Let be a finietly generated -module. Then is a small primary module iff is a small primary module, where Proof: Let , such that , and suppose that So [7]. Then for each On the other hand , , so such that t But is a submodule of (x) and which implies that 0 ( [8]. On the other hand , is small primary , so (0) is a small primary of . Then for some , therefore = . But X is finietly generated, so [13]. Hence . Thus, ) is a small primary module. It follows similarly. Theorem (3.23): Let be a multiplication finietly generated faithful -module. Then is a small primary module iff is a small primary Proof: Suppose that is a small ideal of . But is a multiplication finietly generated faithful, so is a small submodule of X and 0 AX. Since X is small primary and faithful, then 0 = √ √ . But √ √ , therefore √ = 0. Hence R is a small primary ring. Suppose that . So [8]. But X is a multiplication, so [9] . Hence . But R is a small primary ring, so √ =0. Since X is faithful , so √ √ and hence √ = 0 . Thus, √ √ . Therefore X is small primary . Corollary (3.24): Let be a multiplication cyclic faithful -module.Then is a small primary module iff is a small primary Recall that an R-module X is called a scalar module if such that [ 14]. Proposition(3.25): Suppose that is a finietly generated multiplication -module, then is a small primary module iff is a small primary (where Proof: Let be a small S-submodule of X . Then is a small R-submodule of X. Assume that √ and √ . Since is a multiplication finietly generated, hence is a scalar R-module [14]. Hence Thus, and so √ √ . Hence , so , which is a contradiction. Therefore √ √ . Thus, is a small primary . Suppose that and √ √ , so √ and √ . Thus, for some . Define by Clearly, is Rhomomorphism and well-defined. Since , so √ √ (since X is a small primary S-module). Hence , so hich is a contradication. Thus, √ √ and so is a small primary R-maodule.

Proposition(3.26):
If is a scalar -module and is a prime ideal of R, then is a small primary Proof: Since is a scalar -module and is a prime, so =S is a small prime , by [7]. Hence is a small primary Theorem (3.27): Let be a scalar faithful -module . Then is a small primary ff R is a small primary ring. Proof: Since is a scalar, then [15]. But X is faithful, so R Therefore R is a small primary iff is a small primary Theorem (3.28): The followingss are equivalent for a multiplication faithful finietly generatedmodule a-A module is small primary. c-is small primary c-is a small primary Proof: (1) ; by Theorem (3.23). (2) ; since X is a multiplication finietly generated, then X is a scalar, by [14]. Hence, by Theorem (3.27), the result follows.