The soft Jacobson radical of a commutative ring

. In this paper, the notion of the soft Jacobson radical of a ring is deﬁned. A relationship between the soft Jacobson radical of a ring and Jacobson semisimple ring is established. Some properties of this notion have been studied under homo-morphism.


Introduction
The idea of a soft set was introduced by D. Molodtsov [14] in 1999 as a parametrized mathematical tool which has many applications in medical sciences, engineering, economics, etc. Maji and Roy [12] presented a beautiful problem where soft sets are applied in decision making. Applications of soft sets in algebra have been studied rapidly in the last two decades.
Algebraic structure in soft set theory was introduced by Aktas and Cagman [2] in 2007. They defined a soft group as a parametrized family of subgroups of the given group. Extending the notion of a soft group, several algebraic structures like soft ring [1], soft ideal [6], soft vector space [15] etc., have been introduced. In 2012, Cagman et al. [5] defined group structure on a soft set in a new way using set inclusion relation. This concept has been named as soft int-group. Already many researchers have spent their times to extend the notion of soft int-group for further development of soft set theory in the same direction. As a result, the notions of soft int-ring [7], soft int-field, soft int-module [4], soft int-ideal [7] etc, have been established. The soft radical of a soft int-ideal is defined in [9]. This paper focuses on introducing the concept of the soft Jacobson radical of a ring. We define the soft Jacobson radical of a commutative ring R with unity as the soft intersection of all soft maximal int-ideals of R. A connection between the soft Jacobson radical of a ring and Jacobson semisimple ring has been studied. It is established that the homomorphic image of the soft Jacobson radical of a ring R is equal to the soft Jacobson radical of a homomorphic image of R under certain condition. If f is an epimorphism from a ring R to a ring R , then we prove that the homomorphic pre-image of the soft Jacobson radical of R is equal to the soft Jacobson radical of R under a suitable condition.

Preliminaries
We include some basic definitions and results of soft set theory which will be useful in the next section. Throughout this paper, unless otherwise is stated, U is the initial universe, E is the set of parameters, P (U ) is the power set of U , and A ⊆ E.
Let F be a mapping given by F : A → P (A). A pair (F, A) is called a soft set of A over U . When no confusions regarding the parameter set A and the universal set U arise, the soft set (F, A) is simply denoted by F . The collection of all soft sets with parameter set A over U will be denoted by S(A, U ).
Let F, G ∈ S(A, U ). If F (t) ⊆ G(t) for all t ∈ A, then F is called a soft subset of G and is denoted by F ⊆G. Here G is called a soft superset of F and it is denoted by G ⊇F . We write Let F ∈ S(A, U ) and K ⊆ U , the set F K = {x ∈ A : F (x) ⊇ K} is called K-inclusion subset of the soft set F .
The soft intersection F ∩G of two soft sets F and G is defined by It is not difficult to see that the following statement holds true.
Proposition 1 Let F, G ∈ S(A, U ) and K ⊆ U . Then (F ∩G) K = F K ∩G K .
Let A and A be some parameter sets and let f : A → A be any mapping.
for all y ∈ A . Let A, A ⊆ E, G ∈ S(A , U ) and let f : A → A be an onto mapping.
Proposition 2 [7] Let F be a soft int-ring or soft int-ideal of a ring R.
We denote the set of all soft cosets of a soft int-ideal F of a ring R by R F .
The following statement is easy to be verified. Theorem 4 [13] The quotient ring R/JR(R) is a Jacobson semisimple ring.

Soft Jacobson Radical of a Ring
Throughout this section, R is a commutative ring with unity 1. In ring theory [13], the Jacobson radical JR(R) of the ring R is defined as the intersection of all maximal ideals of R. Here, we introduce the following notion.
The soft Jacobson radical SJR(R) of the ring R is defined by Theorem 5 [13] Let y ∈ R. Then y ∈ JR(R) if and only if 1 − xy is a unit in R for all x ∈ R.
Theorem 6 Let y ∈ R and F = SJR(R). Then y ∈ F K with K = F (0) if and only if 1 − xy is a unit in R for all x ∈ R.
Proof. Let y ∈ F K for K = F (0). Suppose 1 − xy is not a unit in R for some x ∈ R. Then, by the crisp concept, there exists a maximal ideal I of R such that 1 − xy ∈ I. Define a soft int-ideal H of R over U by Hence, H is a soft maximal int-ideal of R. Now, y ∈ F K implies F (y) ⊇ K. Since H is a soft maximal int-ideal of R and F = SJR(R), we get H(y) ⊇ F (y). Thus, H(y) ⊇ K ⊃ L. This implies H(y) = N , and hence, y ∈ I. Then xy ∈ I. Therefore, 1 = 1 − xy + xy ∈ I. This implies that I is not a maximal ideal of R, which is a contradiction. Therefore, 1 − xy is a unit in R for all x ∈ R. Conversely, assume that 1 − xy is a unit in R for all x ∈ R. Then by Theorem 5, y ∈ JR(R). Hence, y ∈ M for any maximal ideal M of R. Now, Therefore, y ∈ (F i ) K for all i. By Proposition 1, Theorem 7 Let F = SJR(R). Then r ∈ R is a unit in R if and only if F r is a unit in R F .
Proof. Let r ∈ R be a unit in R. Then there exists s ∈ R such that rs = 1. Hence, F rs = F 1 , where F 1 is the unity element in R F (by Theorem 1). This implies F r F s = F 1 . Thus, F r is a unit in R F . Conversely, let F r be a unit in R F for r ∈ R. Then there exists F s ∈ R F such that F r F s = F 1 . This implies F rs = F 1 , and therefore, Let I be any maximal ideal of R. Define a soft int-ideal H of R by Proof. To prove that the ring R F is a Jacobson semisimple ring, we have to prove JR(R F ) = {F 0 }. Let F r ∈ JR(R F ). Then by Theorem 5, By Theorem 1, F (1−xr)y = F 1 . Hence, F (1−xr)y (a) = F 1 (a) for all a ∈ R. Then F (a − y + xry) = F (a − 1), a ∈ R. Therefore, Let I be any maximal ideal of R. We define a soft int-ideal H of R by where L ⊂ F (0). Then H is a soft maximal int-ideal of R and H(0) = F (0). Since F is the soft Jacobson radical of R, Equation (1) implies By Proposition 2, H(1 − y + xry) = H(0). Therefore, 1 − y + xry ∈ I for any maximal ideal I of R. This implies 1 − y + xry ∈ JR(R). Then by Theorem 5, 1 − (1 − y + xry) = y − xry is a unit in R. This implies that 1 − xr is a unit in R, and hence, r ∈ JR(R).
Suppose {G i : i ∈ Γ} is a collection of all soft maximal int-ideals of R, where Γ is the index set. Then (G i ) L i is the maximal ideal of R, where L i = G i (0). From r ∈ JR(R) it follows that r ∈ (G i ) L i . Hence, G i (r) = L i = G i (0), and therefore, By Proposition 3, we have F r = F 0 . Thus, JR(R F ) = {F 0 }. Therefore, R F is a Jacobson semisimple ring.
Now, for any y ∈ A, using the fact that Since F 1 and F 2 are f -invariant, further we can write .
The following proposition is a generalization of Proposition 7 for an arbitrary soft intersection.
Theorem 9 Let f : R → R be a homomorphism, where R, R are two commutative rings with unity. If each soft maximal int-ideal of R is finvariant, then f (SJR(R)) = SJR(f (R)).
Proof. Let {F i : i ∈ Γ} be the complete collection of soft maximal int-ideals of R, where Γ is an arbitrary non-empty index set. Then by the definition of soft Jacobson radical, we have SJR(R) = i∈Γ F i . Let F i be f -invariant for all i ∈ Γ. Then by Proposition 8, we have Suppose G is a soft maximal int-ideal of f (R). Then by Theorem 3, Theorem 10 Let f : R → R be an epimorphism, where R, R are two commutative rings with unity. Then f −1 (SJR(R )) ⊇SJR(R). Moreover, if each soft maximal int-ideal of R is f -invariant, then f −1 (SJR(R )) = SJR(R).
Proof. Let {G j : j ∈ ∆} be the complete collection of all soft maximal int-ideals of R . Hence, by the definition of soft Jacobson radical, we have SJR(R ) = j∈∆ G j . By Proposition 5, By Theorem 3, f −1 (G j ) is an f -invariant soft maximal int-ideal of R for all j ∈ ∆. There may be some soft maximal int-ideal of R that are not f -invariant. Hence, Therefore, f −1 (SJR(R )) ⊇SJR(R). Now, we assume that each soft maximal int-ideal of R is f -invariant. Let F be a soft maximal int-ideal of R. Then by Theorem 2, f (F ) is a soft maximal int-ideal R and by the Proposition 6, f −1 (f (F )) = F . Hence, for each soft maximal int-ideal F of R, there exist soft maximal int-ideal f (F ) of R such that f −1 (f (F )) = F . Thus, from Equation (2), we have j∈∆ f −1 (G j ) = SJR(R). Therefore, f −1 (SJR(R )) = SJR(R).