A fuzzy characterization of QF rings
© Li et al.; licensee Springer. 2014
Received: 22 February 2014
Accepted: 28 April 2014
Published: 13 May 2014
Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and is an injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy characterizations of right artinian rings and right CF rings. These results throw new light on the research of QF rings and the related CF conjecture.
Recall that a fuzzy subset of a nonempty set X is a map f from X into the closed interval . The notion of fuzzy subset of a set was firstly introduced by Zadeh . Then this important ideal has been applied to various algebraic structures such as groups and rings and so on (see [2–9] etc.). In this article, we introduce some special fuzzy subsets of modules to characterize quasi-Frobenius (QF) rings.
QF rings were introduced by Nakayama  as generalizations of group algebras of a finite group over a field. A ring R is called quasi-Frobenius (QF) if the right R-module is both artinian and injective. QF rings became an important algebraic structure because of their beautiful characterizations and nice applications (see [11–16] etc.). For example, a ring R is QF if and only if every right R-module can be embedded into a free right R-module. Many results of QF rings have been applied into coding theory. During the progress of research on QF rings, many important conjectures arose. One of them is the CF conjecture (see [17, 18] etc.). It says that every right CF ring is right artinian. Recall that a ring R is called right CF if every cyclic right R-module can be embedded into a free right R-module.
Firstly, we introduce the fuzzy homomorphism and weak fuzzy homomorphism of R-modules in Section 2. Then in Section 3, we use weak fuzzy homomorphisms to give a characterization of injective right R-modules. We also obtain some new fuzzy characterizations of right artinian rings. In Section 4, we give a fuzzy characterization of right CF rings. We also give an approach to the CF conjecture through fuzzy viewpoints. Then based on the results we have obtained, we finally get a fuzzy characterization of QF rings.
2 Definitions and examples
Throughout the paper, R is an associative ring with identity and all modules are unitary. For a subset X of a ring R, the right annihilator of X in R is . We write to indicate that M is a right R-module. Let and be two right R-modules. denotes the set of all right R-module homomorphisms from to . means the Cartesian cross product of two sets A and B. We use to denote the image of a map f. For much more notations one is referred to .
, such that ;
, , and implies ;
- (3), ,
, , , .
Definition 2.1 If f satisfies (1), (2), (3), and (4) of the above conditions, f is called a fuzzy homomorphism from to . If f satisfies (1′), (2), (3), and (4) of the above conditions, f is called a weak fuzzy homomorphism from to . We will use (resp., ) to denote the set of all fuzzy homomorphisms (resp., weak fuzzy homomorphisms) from to . It is clear that .
Hence satisfies the condition (3). For the condition (4), let , , . If , it is clear that . If and , then . So . Hence . Thus, . □
According to the condition (3), , , . According to the conditions (3) and (4), , , .
Set and . If is not empty, according to the conditions (3) and (4), is a right R-submodule of .
Let be a submodule of such that . Then by the conditions (1) and (2), for each , there exists a unique such that . Now define a map with . Again by the conditions (2), (3), and (4), it is not difficult to see that .
Definition 2.4 Let , we say if for all , . is defined by , , . It is easy to prove that if , then . It is also clear that and .
Definition 2.5 A weak fuzzy homomorphism is said to be extendable if there exists such that .
Proof It is obvious that f satisfies the conditions (1) and (2).
For the condition (3), let with , where . We only need to consider the following three cases.
Case 1: . It is clear that .
Case 2: . We can suppose that or 1, and . Then and or N, . So . Hence .
Case 3: . Then and . So .
From the above three cases, it is clear that f satisfies the condition (3).
Finally, let with , where . If , then . If for a positive integer k, then . So . Hence . If , then . So . Thus . Therefore, for all , and . Then f satisfies the condition (4). □
Definition 2.7 A weak fuzzy homomorphism is said to be bounded if there exists such that, and , or .
Then . In particular, f is extendable and not bounded.
It is clear that and . So f is extendable. □
3 Fuzzy characterizations of injective modules and artinian rings
According to Baer’s Criterion, a right R-module is said to be injective if every homomorphism from a right ideal I of R to can be extended to a homomorphism from to .
Theorem 3.1 Let R be a ring and a right R-module. Then M is injective if and only if every is extendable.
By a similar proof of Example 2.2, . Since f is extendable, there exists some such that . Now define via , where . According to Remark 2.3(iii), . It is easy to see that . This shows that M is an injective right R-module.
(⇒) Assume that M is an injective right R-module and . Set . By Remark 2.3(ii), I is a right ideal of R. Now define via , where . According to Remark 2.3(iii), . Since M is injective as a right R-module, u can be extended to a homomorphism v from to . By Example 2.2, we have . It is clear that . So f is extendable. □
Next we will give some new fuzzy characterizations of right artinian rings. Recall that a fuzzy subset μ of a ring R is called a fuzzy left (right) ideal of R if μ satisfies: (i) ; (ii) for all . A fuzzy subset f is called finite valued if Imf is a finite set. If Imf is an infinite set, f is called infinite valued.
R is right artinian.
Every fuzzy right ideal of R is finite valued.
For every , f is finite valued.
For every , f is finite valued.
- (b)⇒ (c) Suppose . We can define a fuzzy subset μ of R by
⇒ (d) is obvious.
- (d)⇒ (a) Assume that R is not right artinian. Let be a descending chain of right ideals of R and . Define a fuzzy subset f of by
By Example 2.6, . But f is infinite valued. This is a contradiction. So R is right artinian. □
4 Fuzzy characterizations of right CF rings and QF rings
In this section, we will firstly give a fuzzy characterization of right CF ring. It is well known that a ring R is right CF if and only if for every right ideal I of R, there exist such that (see [, Lemma 7.2]).
Theorem 4.1 A ring R is a right CF ring if and only if for every bounded and extendable , there exist such that .
If , according to Remark 2.3(i), there exists , such that . As , by the condition (4), . Then by the condition (2), we have for all . Since , for all . This implies that . Therefore, . If , let , then . So satisfies the condition (1).
If and with and , then . If and with and , then . So satisfies the condition (2).
Let with , where .
If , then . If , we only need to discuss the following three cases.
Case 1: . Then and .
Case 2: and . Then , and .
So . Since , . Then . Thus, .
Case 3: and . Then and .
So and . If , then because . Thus, . If , then . So .
For all with , where . If , . If , we consider the following two cases.
Case 1: . Then and . Thus, and . So .
Case 2: . Then and . Since , . Then . Thus, . If then . So .
Hence satisfies the condition (4).
From the above, for .
Next we show that .
Case 1: . If , then , . So . By (a) in the above proof, . If , then . So , . Hence for every .
Case 2: . By the definition of I, , . If for some , then . Thus, . If for any , since , . So , . Hence . This is a contradiction.
It is easy to see that and it is bounded and extendable. So there exist such that . Then there exists such that , . By the condition (4), for each , , . But for every , . By the condition (2), , . Hence . This shows that . On the contrary, if , then . So . Hence . This shows that . Therefore, . So R is a right CF ring. □
The following proposition can be looked on as an approach to the CF conjecture.
Proposition 4.2 Let R be a ring. If for every extendable , there exist such that . Then R is right artinian.
Proof By Theorem 4.1, R is a right CF ring. It is well known that a right CF and right noetherian ring is right artinian. So we only need to prove that R is right noetherian.
By Example 2.8, . f is extendable and not bounded. So there exist some such that . According to the condition (1), there exist such that , . Then by the condition (4), . So or for all . Thus, is bounded. This is a contradiction. So R is right noetherian. Thus, R is right artinian. □
Remark 4.3 According to Theorem 3.2, Theorem 4.1 and Proposition 4.2, the CF conjecture is equivalent to saying that every extendable of a right CF ring R is bounded.
At last, we obtain a fuzzy characterization of QF rings.
Theorem 4.4 R is a QF ring if and only if, for every , there exist such that .
Proof Suppose R is a QF ring. Then R is right artinian and the right R-module is injective. By Theorem 3.2, every is finite valued. So f is bounded. Since is injective, by Theorem 3.1, f is extendable. As QF rings are right CF rings, by Theorem 4.1, there exist such that .
Conversely, if for every , there exist , such that . By Proposition 4.2, R is right artinian. According to Definition 2.4, . Then by Theorem 3.1, is injective. So R is a QF ring. □
The authors would like to thank the referees for their nice suggestions and comments. It is supported by NSFC (No. 11371089), NSF of Jiangsu Province (No. BK20130599), NSF of Anhui Province (No. 1408085MA04), the Project-sponsored by SRF for ROCS, SEM and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020).
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