ON LEFT BIDERIVATIONS IN SEMIPRIME SEMIRING

Reverse and Jordan ( , ) − biderivation on Prime and Semi-prime Rings, 2019

In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.

World Scientific

Let R be a ring with centre Z(R). A biadditive symmetric mapping D(., .)

The purpose of this paper is to prove some results concerning symmetric biderivations and symmetric generalized biderivations on prime and semiprime rings which partially extend some results contained in Vukman, J., Symmetric biderivations on prime and semiprime rings

Matematicki Vesnik

The purpose of this paper is to prove some results concerning symmetric biderivations and symmetric generalized biderivations on prime and semiprime rings which partially extend some results contained in

Let S be a semiprime semiring. An additive mapping is called a semi derivation if there exists a function such that (i) , (ii) hold for all . In this paper we try to generalize some properties of prime rings with derivations to semiprime semirings with semiderivations.

Journal of the Australian Mathematical Society, 1980

It is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y e R, there exists a positive integer n depending on v and y such that (\_v)*-x*>'*is central for k = n,n+ 1,H + 2, then R is commutative, thus generalizing a result of Kaya.

Maǧallaẗ ǧāmiʻaẗ kirkūk, 2008

In this paper, two new algebraic structures are introduced which we call a centrally semiprime ring and a centrally semiprime right near-ring, and we look for those conditions which make centrally semiprime rings as commutative rings, so that several results are proved, also we extend some properties of semiprime rings and semiprime right near-rings to centrally semiprime rings and centrally semiprime right near-rings.

Bulletin of the Australian Mathematical Society, 1986

It is shown that if R is a semi prime ring in which (xy)2 -xy is central for every x, y ε R, then R is commutative.

2019

Bu calismada, asal halka uzerinde tanimli sifirdan farkli bir ters (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu ispatlanmistir. Ayrica, 𝑐ℎ𝑎𝑟(𝑅)≠2 olacak bicimdeki degismeli olmayan bir yari-asal 𝑅 halkasi uzerinde tanimli Jordan (𝛼,𝛽)− biturevin ayni zamanda (𝛼,𝛽)− biturev oldugu gosterilmistir. Bunlarin yaninda, Jordan sol (𝛼,𝛼)−biturevli asal halkalarin degismeli olma ozellikleri arastirilmistir.

Boletim da Sociedade Paranaense de Matemática, 2015

Let R be a ring with centre Z(R). A mapping D(., .) : R× R −→ R issaid to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ Rdefined by f(x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious thatin the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive(i.e. additive in both arguments), the trace f of D satisfies the relation f(x + y) =f(x) + f(y) + 2D(x, y), for all x, y ∈ R. In this paper we prove that a nonzero left idealL of a 2-torsion free semiprime ring R is central if it satisfies any one of the followingproperties: (i) f(xy) ∓ [x, y] ∈ Z(R), (ii) f(xy) ∓ [y, x] ∈ Z(R), (iii) f(xy) ∓ xy ∈Z(R), (iv) f(xy)∓yx ∈ Z(R), (v) f([x, y])∓[x, y] ∈ Z(R), (vi) f([x, y])∓[y, x] ∈ Z(R),(vii) f([x, y])∓xy ∈ Z(R), (viii) f([x, y])∓yx ∈ Z(R), (ix) f(xy)∓f(x)∓[x, y] ∈ Z(R),(x) f(xy)∓f(y)∓[x, y] ∈ Z(R), (xi) f([x, y])∓f(x)∓[x, y] ∈ Z(R), (xii) f([x, y])∓f(y)∓[x, y] ∈ Z(R), (xiii) f([x, y])∓f(xy)∓[x, y] ∈ Z(R), (xiv) f([x, y])∓f(xy)∓[y, x] ...