A Determinant Criterion of Invertible Elements in Group Algebras

An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.

Characterization of WAP2 gene in Aegilops tauschii and comparison with homoeologous loci in wheat

The Q/q gene, also known as WAP2, is an important gene for wheat domestication and is a member of the AP2 （APETALA2） class of transcription factors. In the present study, we first isolated the WRAP2 allele （where the superscript ＂t＂ refers to the speciese source, in this case ＂tauschii＂） on chromosome 5D from Aegilops tauschii Coss., the D-genome donor species of common wheat. We found that WRAP2 and the AP2 gene from Arabidopsis share a central core of the AP2 polypeptide, a highly basic 10-amino acid domain, and an AASSGF box, although there are many differences in the 37-amino acid serine-rich acidic domain and the remaining regions. In addition, WRAP2 was highly homologous to the homoeologous loci on 5A and 5B of wheat at both the nucleotide and amino acid level. However, there were some variations that are probably related to gene function. In the first AP2 domain, the amino acids VYL on the 5D and 5A loci were replaced with LLR on 5B. In the 37-amino acid serine-rich acidic domain, WRAP2 on 5D had an extra amino acid insertion. There was also a variation at the 329 amino acid position, which is thought to be related to the appearance of free-threshing wheat. At this position, the amino acid is isoleucine on 5A for the Q allele and valine for the q allele, whereas the amino acid is leucine on 5D and 5B. Furthermore, a Stowaway miniature terminal inverted repeat element （MITE） insertion was present in the ninth intron of WAP2 on 5B of all common wheats and partial tetraploid Triticum turgidum wheats. These results provide new clues for studies into the evolutionary biology of WAP2 and the origin of common wheat.

Differentiation of a Miniature Inverted Transposable Element （MITE） System in Asian Rice Cultivars and Its Inference for a Diphyletic Origin of Two Subspecies of Asian Cultivated Rice

In the present study, we report a survey on a Miniature Inverted Transposable Element （MITE） system known as mPing in 102 varieties of Asian cultivated rice （Oryza sativa L.）. We found that mPing populations could be generalized Into two families, mPing-1 and mPing-2, according to their sequence structures. Further analysis showed that these two families of mPing had significant bias in their distribution pattern in two subspecies of rice, namely O. sativa ssp. japonica and indica. 0. sativa japonica has a higher proportion of mPing-1 as a general trait, whereas 0. sativa indica has a higher proportion of roPing-2. We also examined the mPing system In a doubled haploid （DH） cross-breeding population of jingxi 17 （japonica） and zhaiyeqing 8 （indica） varieties and observed that the mPing system was not tightly linked to major subspecies-determining genes. Furthermore, we checked the mPing system in 28 accessions of Asian common wild rice O. rufipogon and found the roPing system in 0. rufipogon. The distribution pattern of the roPing system in O. rufipogon indicated a diphyletlc origin of the Asian cultivated rice O. sativa species. We did not find the mPing system in another 20 Oryza species. These results substantiated a previous hypothesis that O. ruflpogon and O. nivara species were the closest relatives of O. sativa and that the two extant subspecies of O. sativa were evolved independently from corresponding ecotypes of O. ruflpogon.

The Comaximal Graphs of Noncommutative Rings

For a ring R(not necessarily commutative)with identity,the comaximal graph of R,denoted byΩ(R),is a graph whose vertices are all the nonunit elements of R,and two distinct vertices a and b are adjacent if and only if Ra+Rb=R.In this paper we consider a subgraphΩ_(1)(R)ofΩ(R)induced by R\Uℓ(R),where Uℓ(R)is the set of all left-invertible elements of R.We characterize those rings R for whichΩ_(1)(R)\J(R)is a complete graph or a star graph,where J(R)is the Jacobson radical of R.We investigate the clique number and the chromatic number of the graphΩ_(1)(R)\J(R),and we prove that if every left ideal of R is symmetric,then this graph is connected and its diameter is at most 3.Moreover,we completely characterize the diameter ofΩ_(1)(R)\J(R).We also investigate the properties of R whenΩ_(1)(R)is a split graph.

On the Arithmetic of Endomorphism Ring End (Zp×Zpm)

For a prime p,let E_（p,p^m）={（a p^（m-1） b d）｜a,b,c∈Z_p,d∈Z_（p^m）}. We first establish a ring isomorphism from Z_（p,p^m） onto E_（p,p^m）. Then we provide a way to compute-d and d^（-1） by using arithmetic in Z_p and Z_（p^m）, and characterize the invertible elements of E_（p,p^m）. Moreover, we introduce the minimal polynomial for each element in E_（p,p^m） and give its applications.