Topics in the isomorphism of group rings

Introduction: Let R be a ring and let G be a group. The group ring R(G) of G over R is the free left R-module over the set of elements of G as a basis in which the multiplication induced by G is extended linearly to R(G) , [12]. A twisted group ring Ry (G) of G over R is an R-algebra with basis {ḡ |g ε G} and with an associative multiplication ḡ ħ = Y(g, h) ḡħ for all g, h ε G , where y(g, h) is a unit in the centre of R , [13]. In [5] Higman proved that the only units of finite order in the group ring R(G), where R is the ring of rational integers and G is a finite abelian group, are ± g, g ε G . In [16] Sehgal proved that the only units of finite order in the group ring R(G), where R is the ring of rational integers and G is an arbitrary abelian group, are ± t where t is a torsion element of G. Moreover in [16] he proved that the units of R(G), where R is an integral domain and G is a torsion-free abelian group, are of the form r g where r is a unit in R and g ε G. Also in [15] he proved that the units of R(), where R is a commutative ring with no non-zero nilpotents and no non-trivial idempotents and is an infinite cyclic group, are of the form r g where r is a unit in R and g ε . In [17] Zariski and Samuel studied R-automorphisms of the polynomial rings R[x], (that is, automorphisms of R[x] which restrict to the identity mapping on R) where R is an integral domain. In [3] Gilmer determined R-automorphisms of the polynomial rings R[x] where R is a commutative ring. In [2] Coleman and Enochs studied the corresponding results in general. In [9] Parmenter studied R-automorphisms of the group ring R() where is an infinite cyclic group and he determined necessary and sufficient conditions that x → Σ aixi induces an R-automorphism of R(). He also studied the units of R(G) where R is a commutative ring and G is a right-ordered group. This thesis consists of five chapters. Chapter 1 contains some well known results and definitions that are needed in this thesis. In Chapter 2 we extend some ideas of [9] to a twisted group ring RY () where is an infinite cyclic group and we determine a necessary and sufficient condition that x̄ → Σ l aixi induces an R-automorphism of RY () . Chapter 3 studies R-automorphism of R(G) where R is either a field or a ring with a unique proper ideal and G is a finitely generated torsion-free abelian group. In Chapter 4 we determine the units and study the K-automorphisms of K( x ) where K is a field and is an infinite 2 cyclic group, y2 = 1. In [10] Passman proved that the group algebras of all non-isomorphic p-groups of order at most p4 over the prime field of p elements are non-isomorphic. In Chapter 5 we attempt to find the corresponding results for the p-groups of order p 5, but the problem is still open

Mazdek hareketi üzerine bir araştırma

Bu makalenin Farsça aslı “Berresî Cunbiş-i Mazdek” adıyla “Pejûheşnâme-i Ulûm-i İnsânî (Şehit Beheşti Üniversitesi Edebiyat ve Beşeri Bilimler Fakültesi Dergisi) 27 (1379), s. 108-126.”de yayımlanmıştır.Eski tarihçilerin İslamiyet öncesi İran tarihi hakkında Farsça veya Arapça nakilleri önemli bir kaynağa, yani Sâsânî saltanatının sonunda (muhtemelen III. Yezdicerd döneminde) Pehlevî dilinde kaleme alınmış yarı resmi bir metin olan Ħudâynâme’ye dayanır. Onun Arapça çevirileri arasında en bilineni İbnü’l-Mukaffa’nın tercümesidir. İbnü’l-Mukaffa ve Pehlevî Ħudâynâme’nin diğer yazarları, Sâsânî saltanat yıllıkları veya mezkûr yıllıklara dayanan başka kaynaklardan faydalanmışlardır. Ħudâynâme’de olayların gidişatı hükümdarların, soyluların ve Zerdüştî din adamlarının bakış açısıyla ifade edilmektedir. Bu bakış açısı Müslüman tarihçi ve yazarların metinlerine doğrudan yansımıştır. İslam tarihçileri sayılı birkaç örnek haricinde her ne kadar menşeini ve kaynaklarını belirtmemiş olsalar da yazdıklarını Ħudâynâme’nin Arapça çevirisinden alıntılamış oldukları açıktır

Floristic, life form and chorological studies of the Abshar protected area, Shirgah, Mazandaran Province, north of Iran

The Abshar protected area with 3639 ha and an altitude ranging from 400-855 m a.s.l. is one of the forest areas that due to its topography is covered by Carpinus betulus, Parrotia persica and Diospyros lotus speices. The floristic study of this area is long with sampling plots. The floristic-physiognomic investigation showed that flora of this region included 99 plant species which belonged to 81 genera and 49 families. The largest family was Rosaceae with 10 species. Classification based on life form spectrum indicated that geophytes (31.31%), phanerophytes (29.29%), hemicryptophytes (25.25%) and therophytes (14.14%) comprise the plants in the studied area. The results of chorological studies showed that the chorotype form Euro-Siberian elements (32 taxa, 32.32%) were the most important phytochorion in this area. Long period of wetness during the growing season and relatively high annual precipitation are the reasons of the high proportion of geophytes existence in the studied area. Phytogeographical comparison of the Abshar protected forest and the other forests in north of Iran identified two peaks in phytochoria curves, one in Euro-Siberian and the other one is in the Pluriregional elements

A floristic study of the Sorkhankol Wildlife Refuge, Guilan province, Iran

Sorkhankol Wildlife Refuge with an area of 1209 ha is located in the central part of Anzali Wetland. In total, 81 species belonging to 35 families and 68 genera were surveyed and identified on the basis of a floristic study from July 2013 through June 2014. The largest families are Poaceae (11 taxa), Asteraceae (8 taxa) Apiaceae, Brassicaceae and Cyperacae (5 taxa). The dominant life forms were cryptophyte (43.21%), followed by the therophytes (39.51%), hemicryptophytes (13.58%) and phanerophytes (3.7%). From the chorological point of view, the largest proportion of the flora belongs to the pluriregional elements (44.44%). A comparison between our study and other parts of the Anzali Wetland showed that Sorkhankol was not particularly species-rich. Currently, the major threats to the research area include eutrophication, pressure from boating and fishing activities, invasion of exotic species and other human induced disturbances