A Study of Applications and their Involution of left ∗-Centralizers and ∗-Mappings in Rings

An additive mapping T : R → R is called a left centralizer in case T(xy) = T(x)y holds for all x, y ∈ R. An additive mapping T : R → R is said to be a reverse left centralizer if T(xy) = T(y)x holds for all x, y ∈ R. The definition of a reverse right centralizer should be self-explanatory. For a semiprime ring R, all left centralizers are of the form T(x) = qx for all x ∈ R, where q is an element of Martindale right ring of quotients Qr(R) of R for details). In case if R has an identity element, then T: R → R is a left centralizer if and only if T is of the form T(x) = ax for all x ∈ R and some fixed element a ∈ R. In case T : R → R is a two-sided centralizer, where R is a semiprime ring with extended centoid C, then there exists an element λ ∈ C such that T(x) = λx for all x ∈ R. An additive mapping T : R → R is called a Jordan left centralizer (resp. Jordan right centralizer) if T(x 2 ) = T(x)x (resp. T(x 2 ) = xT(x)) holds for all x ∈ R. Clearly, every left centralizer (resp. right centralizer) on a ring R is a Jordan left centralizer on R. But the converse of this statement need not be true in general. However, in case of prime ring of characteristic different from two both concepts coincide. Further, Zalar proved this result in the setting of semiprime ring of characteristic different from two. More related results on centralizers in rings and algebras can be looked where further references can be found. Let R be a ring with involution ∗. Motivated by the definitions of left (resp. right) centralizer and Jordan left (resp. right) centralizer in rings, Ali and Foˇsner introduced the notion of left (resp. right) ∗-centralizer and Jordan left (resp. right) ∗- centralizer as follows: an additive mapping T : R −→ R is said to be a left ∗-centralizer (resp. Jordan left ∗-centralizer) if T(xy) = T(x)y ∗ (resp. T(x 2 ) = T(x)x ∗ ) holds for all x, y ∈ R. The definition of right ∗-centralizer (resp. Jordan right ∗-centralizer) should be self-explanatory. An additive mapping T : R −→ R is said to be a reverse left ∗-centralizer if T(xy) = T(y)x ∗ is fulfilled for all x, y ∈ R. Reverse right ∗-centralizer is defined in a similar way. An additive mapping T : R −→ R is called a ∗-centralizer (resp. reverse ∗-centralizer) if T is both a left and a right ∗-centralizer (resp. reverse left and right ∗-centralizer). Note that for some fixed element a ∈ R, the mapping x 7→ ax∗ is a Jordan left ∗-centralizer and x 7→ x ∗a is a Jordan right ∗-centralizer on R. Clearly, every reverse left ∗-centralizer on a ring R is a Jordan left ∗-centralizer. Thus, it is natural to question that whether the converse of above statement is true. In study is shown that the answer to this question is affirmative if the underlying ∗-ring R is 2-torsion free semiprime ring. Further, we establish a result concerning additive mapping T : R → R satisfying the relation T(x m+n+1) = (x ∗ ) nT(x)(x ∗ ) m for all x ∈ R, where m and n are positive integers. Moreover, some nice

Perhaps, it was Breˇsar and Zalar [49] who first introduced the concept of Jordan centralizers of a ring R, and established that on a prime ring of characteristic different from two every Jordan left centralizer (resp. Jordan right centralizer) is a left centralizer (resp. right centralizer) on R [49, Proposition 2.5]. Further, in [146] Zalar generalized the above mentioned result for semprime ring (without involution). In view of this result, it is natural to question that whether the above result is true in case of ring with involution. In the present study, it is shown that the answer to this question is affirmative if the underlying ∗-ring R is a 2-torsion free semiprime. In fact, we prove the following result: Hence, we obtain S(x 2 ) = xS(x) for all x ∈ R. Thus, S is a Jordan right centralizer on R. In view S is a right centralizer that is, S(xy) = xS(y) for all x, y ∈ R. This implies that (T(xy))∗ = x(T(y))∗ for all x, y ∈ R. By applying involution to the both sides of the last relation, we find that T(xy) = T(y)x ∗ for all x, y ∈ R. This completes the proof of the proposition. Theorem 2. Let R be a non-commutative prime ring with involution ∗ such that char. Then the following conditions are mutually equivalent: (i) R is normal (ii) there exists a nonzero commuting Jordan left ∗-centralizer T- on R. Proof. Suppose R is a normal ring. Then the mapping x 7→ x ∗ is a commuting nonzero Jordan left ∗-centralizer on R. Now suppose (ii) holds, we have to prove R is normal. there exists µ ∈ C and a map ν : R → C such that Since the identity involves involution, so it is a functional identity or the so-called gidentity. In view of Lemma 1.3.2, we conclude that qx∗ − µx ∈ C for all x ∈ Qs(R), the symmetric ring of quotients. Note that Qs(R) has the identity element 1. Replacing x by 1 in the above expression, we see that q − µ ∈ C. This implies that [q, y] = 0 for all y ∈ Qs(R). Thus, The above theorem has the following interesting consequence: Corollary. Let R be a non-commutative prime ring with involution ∗ such that char(R) 6= 2. If T is a nonzero Jordan left ∗-centralizer on R such that [T(x), x] = 0for all x ∈ R, then there exists λ ∈ C, the extended centroid of R such that T(x) = λx∗ for all x ∈ R. Theorem. Let R be a non-commutative prime ring with involution ∗ such that char(R) 6= 2. If T1 and T2 are two nonzero Jordan left ∗-centralizers on R such that T1(x)x − xT2(x) = 0 for all x ∈ R, then R is normal. Proof. By the given hypothesis, we have

During the last few decades, there has been ongoing interest concerning the left centralizers (resp. Jordan left centralizers) on prime and semiprime rings. Recently, many authors viz; have obtained some interesting results in rings and algebras. In Vukman provedthat if R is a non-commutative 2-torsion free semiprime ring and S, T: R → R- are left centralizers such that I n case R is a prime ring and then there exists λ ∈ C such that T = λS (S = λT). The intent of this study is to study similar types of We begin with the following: Lemma 1. Let R be a non-commutative prime ring with involution ∗ and let T : R −→ R be a Jordan left ∗-centralizer on R. If T(x) ∈ Z(R) for all x ∈ R, then T = 0. Proof. By the hypothesis, we have [T(x), y] = 0 for all x, y ∈ R. Substituting x 2 for x in the above relation, we obtain

CONCLUSION

In the present study we give a brief exposition of some important terminology in the theory of rings and algebras. Examples and counter examples are also included in this study to make the matter presented in the study self-explanatory and to give a clear sketch of the various notions. The knowledge of some elementary concepts like groups, rings, ideals, fields, modules, homomorphism etcetera have been presumed. Throughout the thesis, unless otherwise mentioned, R will denote an associative ring (may be without unity) containing at least two elements. Throughout the discussion all rings are associative unless indicated otherwise and Z(R) denotes the commutator) of elements x and y, and x ◦ y is called the Jordan product (or the anti-commutator) of x and y. the commutativity of prime rings with involution. An additive mapping x 7→ x ∗ on a ring R is called an involution on R if (xy) ∗ = y ∗x ∗ and (x ∗ )∗ = x holds for all x, y ∈ R. A ring equipped with an involution is called a ring with involution or ∗-ring. Let d : R → R be an additive mapping such that d(xy) = d(x)y + xd(y) for all x, y ∈ R. Then d is said to be a derivation on R. Let S be a nonempty subset of R. A mapping f : R → R is called centralizing on S if [f(x), x] ∈ Z(R) for all x ∈ S, and is called commuting on S if [f(x), x] = 0 for all x ∈ S. Motivated by the existence of centralizing mappings in rings, the notions of ∗-centralizing and ∗-commuting mappings in rings with involution.

REFERENCES

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Research Scholar, Shri Krishna University, Chhatarpur M.P.