Let E : y 2 = x 3 +Ax 2 +Bx +C be a nonconstant elliptic curve over Q(t) with at least one nontrivial Q(t)-rational 2-torsion point. We describe a method for finding t 0 ∈ Q for which the corresponding specialization homomorphism t → t 0 ∈ Q is injective. The method can be directly extended to elliptic curves over K(t) for a number field K of class number 1, and in principal for arbitrary number field K. One can use this method to calculate the rank of elliptic curves over Q(t) of the form as above, and to prove that given points are free generators. In this paper we illustrate it on some elliptic curves over Q(t) from an article by Mestre.

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Abstract. Let E : y 2 = (x − e 1 )(x − e 2 )(x − e 3 ), be a nonconstant elliptic curve over Q(T ). We give sufficient conditions for a specialization homomorphism to be injective, based on the unique factorization in Z[T ] and Z.The result is applied for calculating exactly the Mordell-Weil group of several elliptic curves over Q(T ) coming from a paper by Rubin and Silverberg.

Abstract. We show that a generalized derivation on a prime ring, that acts as a homomorphism or an anti-homomorphism on a non-zero ideal in the ring, is the zero map or the identity map.Let R be an associative ring, let d be a derivation on R (i.e. an additive function on R satisfying d(xy) = d(x)y + xd(y) for all x, y ∈ R) and let F : R → R be a generalized derivation associated to d (i.e. an additive function satisfying F (xy) = F (x)y + xd(y) for all x, y ∈ R).We say that R is prime if the relation aRb = 0 implies that a = 0 or b = 0, for all a, b ∈ R. Note that if R is a prime ring and I is a non-zero ideal of R, then the relation aIb = 0 implies that a = 0 or b = 0, for all a, b ∈ R In [R, Theorem 1.2] the following statement is stated. Assume that R is 2-torsion free and prime. in R then R is commutative. It seems that the assumptions in this statement are contradictory. Also, despite an ingenious argument the conclusion is incomplete. Using a similar argument we prove the following: Theorem 1. Let R be an associative prime ring, let d be any function on R (not necessary a derivation nor an additive function), let F be any function on R (not necessarily additive) satisfying F (xy) = F (x)y + xd(y) for all x, y ∈ R, and let I be a non-zero ideal in R.2000 Mathematics Subject Classification. 16W25.

We describe decomposition of polynomials f n := f n,B,a defined bywhere B and a are rational numbers. We also present an application to related Diophantine equations.