Let K/F denote an abelian extension of number fields with associated rings of integers OK and OF . Let S denote a finite set of places of F containing the archimedean places and those which ramify in K. Assume that S contains at least one place v that splits completely in K and that |S| ≥ 2. For each ideal n ⊂ OF not divisible by a prime that ramifies in K, we denote by σn the associated Froben...

Generalizing the work of Lubin in the one dimensional case, conditions are found for the existence of canonical subgroups of finite height commutative formal groups of arbitrary dimension over local rings of mixed characteristic (0, p). These are p-torsion subgroups which are optimally close to being kernels of Frobenius homomorphisms. The Fp ranks of the first flat cohomology groups of these c...

The m-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). m-spotty byte error control codes are very suitable for burst correction. M. Özen and V. Siap [7] proved a MacWilliams identity for the m-spotty Rosenbloom-Tsfasman (shortly RT) weight enumerator...

In coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring Zm of integers modulo m and the finite fields. In this paper, we derive an isometric mapping from Z8 to Z4 2 from the composition of the Gray isometries on Z8 and on Z4 . The image under this composition of a Z8-linear block code of length n with homogeneous distance d is a (...

an r-module m is called epi-retractable if every submodule of mr is a homomorphic image of m. it is shown that if r is a right perfect ring, then every projective slightly compressible module mr is epi-retractable. if r is a noetherian ring, then every epi-retractable right r-module has direct sum of uniform submodules. if endomorphism ring of a module mr is von-neumann regular, then m is semi-...

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions canonical bases "theta functions" on the coordinate rings various log Calabi-Yau spaces, including cluster varieties. We prove that theta for varieties are determined by certain descendant Gromov-Witten invariants symplectic leaves mirror/Langlands dual var...

An extension theorem for general weight functions is proved over nite chain rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. 1. Background In her doctoral dissertation, MacWilliams 7], 8] proved an equivalence theorem: two linear codes C 1 ; C 2 F n deened over a nite eld F are equivalent up to monomia...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...