A ring R is called right strong stably finite (r.ssf) if for all n ≥ 1, injective endomorphisms of R (n) R are essential. If R is an r.ssf ring and e is an idempotent of R such that eR is a retractable R-module, then eRe is an r.ssf ring. A direct product of rings is an r.ssf ring if and only if each factor is so. The R.ssf condition is investigated for formal triangular matrix rings. In partic...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

Suppose K is a eld of characteristic 0, Ka is its algebraic closure, p is an odd prime. Suppose, f(x) 2 K[x] is a polynomial of degree n 5 without multiple roots. Let us consider a curve C : y = f(x) and its jacobian J(C). It is known that the ring End(J(C)) of all Ka-endomorphisms of J(C) contains the ring Z[ p] of integers in the pth cyclotomic eld (generated by obvious automorphisms of C). W...

Let f be a continuous ring endomorphism of ZpJxK/Zp of degree p. We prove that if f acts on the tangent space at 0 by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over Zp. The proof relies on finding a stable embedding of ZpJxK in Fontaine’s crystalline period ring with the property that f appears in the monoid of en...

We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the N\'{e}ron-Severi groups. The discrete logarithm problem is reduci...

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them bounded linear operators endomorphisms free modules infinite rank. In particular it is proved that every operator an infinite-dimensional complex Hilbert space a sum four automorphisms order 3 simple ring obtained quotient the endomorphism vector modulo its...