Subrings which are closed with respect to taking the inverse

Let S be a subring of the ring R. We investigate the question of whether S intersected by U(R) is equal to U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation

Crop and climate suitability for irrigated agriculture in the Midlands area of Western Australia, 2nd edition

The Midlands groundwater and land assessment is a $4.7 million Water for Food project. Its aim is to confirm groundwater availability at one or more focus areas that may form precincts of 2000–3000 hectares suitable for intensive irrigated horticulture. This report forms part of this project. Firstly, this report describes the climate of the Midlands study area and highlights the subtle differences between the two selected focus areas, Irwin and Dinner Hill. It discusses the importance of climate in determining crop suitability. We also investigated the following additional factors that determine crop suitability: water quality, water quantity, land capability (soils) and environmental impact. Secondly, this report examines potential commercial horticultural crops for the Midlands area. We found that the combination of climate and the range of soils in the study area would suit a wide range of horticultural crops. The main limiting factor for extensive horticultural development in the study area is the availability of water that is suitable for irrigation. High temperatures, wind and water quality are also important management considerations. The first edition of this report was updated after corrections were made in related reports, specifically resource management technical reports 406 and 408. This second edition contains those corrections, and some outdated mapping has been replaced or removed. Other minor editorial changes have also been made in this edition

Jordan Derivations and Antiderivations of Generalized Matrix Algebras

Let \mathcal{G}=[A & M N & B] be a generalized matrix algebra defined by the Morita context $(A, B,_AM_B,_BN_A, \Phi_{MN}, \Psi_{NM})$. In this article we mainly study the question of whether there exist proper Jordan derivations for the generalized matrix algebra $\mathcal{G}$. It is shown that if one of the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ is nondegenerate, then every antiderivation of $\mathcal{G}$ is zero. Furthermore, if the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ are both zero, then every Jordan derivation of $\mathcal{G}$ is the sum of a derivation and an antiderivation. Several constructive examples and counterexamples are presented.Comment: 15 page

Centralizers in endomorphism rings

We prove that the centralizer Cen(f) in Hom_R(M,M) of a nilpotent endomorphism f of a finitely generated semisimple left R-module M (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m x m matrix algebra M_m(R[z]), where m is the dimension (composition length) of ker(f). If R is a local ring, then we provide an explicit description of the above Cen(f). If in addition Z(R) is a field and R/J(R) is finite dimensional over Z(R), then we give a formula for the Z(R)-dimension of Cen(f). If R is a local ring, f is as above and g is an arbitrary element of Hom_R(M,M), then we give a complete description of the containment Cen(f) in Cen(g) in terms of an appropriate R-generating set of M. Using our results about nilpotent endomorphisms, for an arbitrary (not necessarily nilpotent) linear map f in Hom_K(V,V) of a finite dimensional vector space V over a field K we determine the PI-degree of Cen(f) and give other information about the polynomial identities of Cen(f)