For σ−PWB extensions, we extend to modules the theory of Gröbner bases of left ideals presented in [5]. As an application, if A is a bijective quasi-commutative σ − PWB extension, we compute the module of syzygies of a submodule of the free module A.

Effect of high temperature tolerance in relation to carbon partitioning and grain sink activity was studied in ten genotypes of wheat (viz C 306, PBW 343, PBW 502, PBW 509, PBW 521, PBW 522, PBW 550, Inqalab, WL 711 and Sonalika) under normal (November, 25.6 ̊C during grain filling) and late planting conditions (December, 29.4 ̊C).

An extension L/K of skew fields is called a leftpolynomialextension with polynomial generator 0 if it has a left basis of the form 1, i3, ti’, , Bnm’ for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extens...

The second cohomology group of a left skew brace with coefficients in trivial non-trivial actions is defined, its connection extensions by braces established and Wells' like exact sequence relating the inducible automorphisms an extension constructed.

We investigate the amenability of skew field extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable finitely generated skew fields of infinite Gelfand-Kirillov transcendence degree. AMS Subject Classifications: 12E15, 43A07