The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.
Authors:Muhammad Saad, Helen Mary, Umar Amjid, Ghulam Shabir, Kashif Aslam, Shahid Masood Shah and Abdul Rehman Khan
Tartary buckwheat, known for its rich source of health beneficial secondary metabolites, is cultivated in many areas of the world. Among different environmental factors, photoperiod strongly influence its growth, flowering time, and ultimately the yield. In this context, epigenetics could contribute significantly in the regulation of plant response against changing environment. Therefore, with the aim to study the involvement of DNA methylation in photoperiod mediated plant response, genome-wide DNA methylation analysis was performed in two accessions (A1 and A2) of Tartary buckwheat using three photoperiodic treatments, i.e., 10-hr light/day (T1), 12-hr light/day (T2), and 14-hr light/day (T3). Flowering time and plant fresh weight data revealed that accessions A1 and A2 prefer T1 and T2 treatments, respectively. Total DNA methylation ratio increased with the increase in photoperiod in accession A1 but decreased under same conditions in accession A2. Full methylation increased significantly while intensive decrease in hemimethylation was noted from T2 to T3 in A1, whereas full methylation strongly increased and hemimethylation strongly decreased from T1 to T2 in A2. Overall, the DNA methylation events appeared more frequently than demethylation events. This study reports for the first time an accession-/genotype specific pattern of shift in the DNA methylation under different photoperiodic treatments that will pave the way toward identification of specific genes involved in the regulation of plant response against photoperiodic stress.