توصيفگر ها :
حلقه تميز , حلقه خارج قسمتي , حلقه قوياً تميز , حلقه موضعي , حلقه قوياً m-تميز , حلقه m-تميز , حلقه m-موضعي , عضو m-توان
چكيده فارسي :
حلقه ي R تميز ناميده مي شود، هرگاه هر عضو R ∈ a را بتوان به صورت e + u = a نوشت، كه در آن e^2 = e عضو خودتوان از
R و u عضو يكه از R باشند. در ادامه ي تعريف بالا اگر ue = eu ،آنگاه حلقه ي R را قوياً تميز گويند. اگر در تعاريف بالا داشته
e) 2 ≥ m (در اين صورت e را m-توان گويند و تعاريف حلقه m-تميز و حلقه قوياً m-تميز به دست خواهد آمد.
m = e باشيم
در اين پايان نامه ويژگي هايي از حلقه m-تميز و قوياً m-تميز را معرفي و ثابت مي كنيم كه اگر حلقه R ،m-تميز باشد، آنگاه حلقه
ماتريس (R(M_n هم m-تميز است. همچنين فرض مي كنيم I ايدآلي از حلقه R باشد به طوري كه (R(J ⊆ I و R در خاصيت
جابه جايي به پيمانه I صدق كند. در اين صوت اگر حلقه R/I ،قوياً m-تميز باشد و m-توان ها به هنگ ايدآل I بالا بروند، آنگاه
حلقه R قوياً m-تميز است. به ويژه ثابت مي كنيم يك حلقه قوياً m-تميز است اگر و تنها اگر حلقه خارج قسمتي آن با يك ايدآل
پوچ تحت شرايط خاصي قوياً m-تميز باشد. همچنين مفهوم حلقه m-موضعي را به عنوان زير رده اي از حلقه موضعي تعريف مي كنيم
و ثابت مي كنيم كه يك حلقه m-موضعي است اگر و تنها اگر m-تميز باشد و هيچ عضو m-توان نابديهي نداشته باشد
چكيده انگليسي :
Throughout this paper we consider a ring with identity 1. We denote by Id(R), U(R), J(R) and Mn(R) the set
of idempotent, the set of units, the Jacobson radical and the ring of n × n matrices over R, Recpectivily.
R is said to be m-clean if every element a ∈ R can be written as a = u + e where e
2 = e is an idempotent and u
is a unit element of R. Following the above definition if eu = ue, then R is said strongly m-clean ring.
If we have high definitions e
m = e (m ⩾2), then element e of a ring R is said to be m-potent, so definition m-clean
ring and strongly m-clean ring.
In this paper, we introduce the characterization of a m-clean ring and strongly m-clean ring and prove that the following theorem: Let e be an m-potent element of a ring R such that the subrings e
m−1Rem−1
and (1−e
m−1
)R(1−
e
m−1
) are m-clean.Then R is also m-clean.
In general, we have the following result.
Let {e1, . . . , en} be a set of m-potent elements of a ring R such that e
m−1
i
and e
m−1
j
are mutually orthogonal for
all i ̸= j, where 1 ≤ i, j ≤ n. Suppose that 1 = e1 + e2 + · · · + en and each e
m−1
i Rem−1
i
is m-clean for every
i = 1, 2, · · · , n. Then R is m-clean.
By using the above theorem, we prove that if a ring R is m-clean then Mn(R) is m-clean.
In the following, we show the characteristics of a m-clean ring in terms of a m-local ring and some other notions.
Let R be a ring with “m-unit property” and Char(R) = m. Then the following statements are equivalent.
(i) R is m-local.
(ii) R is m-clean and it has no nontrivial m-potent element.
(iii) R is m-exchange and it has no nontrivial m-potent element.
(iv) R is m-potent and it has no nontrivial m-potent element.
(v) R is m-semipotent and it has no nontrivial m-potent element.
We also suppose that I is an ideal of R such that I ⊆ J(R) and R satisfying in the modulo commutative property.
Then if R/I is strongly m-clean and m-potents lift modulo I, then R is strongly m-clean.
In particular, we prove that a ring is strongly m-clean if and only if its quotient ring by a nil ideal is strongly m-clean
under certain conditions.
We also introduce the notion of m-local ring as a subclass of local ring. we establish that a ring is m-local if and only
if it is m-clean and it has no non-trivial m-potent element
