T-Radicals in the Category of Modules
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T-Radicals in the Category of Modules E. A. TIMOSHENKO Tomsk State University, 35 Mira Prosp., flat 172, 634027 Tomsk, Russia. e-mail: [email protected] Abstract. This paper deals with the class of idempotent radicals defined by means of a tensor product. Their effect on Abelian groups is described. We also establish their connection with attracting modules. Mathematics Subject Classification (2000): 18E40. Key words: Abelian group, module, radical, tensor product, torsion.
This paper investigates the properties of a special class of radicals defined in the category of modules (T-radicals). In a sense, the notion of a T-radical synthesizes the definitions of an E-radical, appearing for the first time in [4], and a T-module [3]. The first section is concerned with modules over some ring S, the second one deals with modules both over the ring S and another ring R. It is suggested that R and S are associative rings with unity, modules are unitary and, unless otherwise stated, right. The category of right S-modules is designated mod-S. The word “group” means the Abelian group. 1. T(F )-radicals In this section, modules over S are dealt with. F designates some fixed left S-module. DEFINITION 1.1. An S-module A is called a T(F )-module if A ⊗S F = 0. The class of all T(F )-modules is denoted by T (F ). The class T (F ) is closed under homomorphic images, extensions and direct sums. In general, it is not closed under direct products. Let us recall some definitions of the theory of radical class. DEFINITION 1.2. We say that in the category mod-S a preradical λ is defined, if to each S-module A is assigned its submodule λ(A), so that for any S-homomorphism ϕ: A → B the relation ϕ(λ(A)) ⊂ λ(B) holds true. Let λ be a preradical. A class of all S-modules A for which λ(A) = A is called a λ-radical class. Let us consider the following (possible) properties of a preradical λ:
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R1. λ(λ(A)) = λ(A) for any A ∈ mod-S. R1∗ . λ(A/λ(A)) = 0 for any A ∈ mod-S. R2. λ(B) = B ∩ λ(A) for any A ∈ mod-S and B ⊂ A. DEFINITION 1.3. A preradical λ is called a radical if R1∗ holds. A radical which satisfies R1 is said to be an idempotent radical. A preradical λ is called a torsion if it satisfies R2 and R1∗ . It is obvious that any torsion is an idempotent radical. WF (A) will designate the sum of all submodules B of A ∈ mod-S such that B is a T(F )-module. WF is an idempotent radical, and T (F ) is its radical class [2]. From here on we will call this idempotent radical simply “T(F )-radical”. DEFINITION 1.4. Let A be an S-module. The F -neutralizer of A is the set of all elements a ∈ A such that for all f ∈ F we have a ⊗S f = 0 in the tensor product A ⊗S F . This neutralizer is denoted by nF (A). The neutralizer nF is a radical. The following equivalences are obvious: WF (A) = A ⇐⇒ A ∈ T (F ) ⇐⇒ A ⊗S F = 0 ⇐⇒ nF (A) = A. In accordance with the definition of the T(F )-radical, we immediately obtain from [2] that WF is the largest idempotent radical λ such that λ(A) ⊂ nF (A) for all A ∈ mod-S. Therefore the in
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