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SUPERVISOR
Mohammad-Reza Vedadi,Hossein Khabazian
محمد رضا ودادی (استاد مشاور) حسین خبازیان اصفهانی (استاد راهنما)
 
STUDENT
Fatemeh Moslehi
فاطمه مصلحی

FACULTY - DEPARTMENT

دانشکده ریاضی
DEGREE
Master of Science (MSc)
YEAR
1386
A ring R with identity is called strongly clean if every element is the sum of an idempotent and a unit which commute. We will completely characterize the commutative local rings for which Mn(R) is strongly clean, in terms of factorization in R[T ]. We also obtain similar elementwise results which show additionally that for any monic polynomial f ? R[T ], the strong cleanness of the com-panion matrix of f is equivalent to the strong cleanness of all matrices with characteristic polynomial f. And the same direction, for example we proof following corollary. Corollary 23.4: For a commutative local ring R and n ? 1, the following are equivalent: (1) Mn(R) is strongly clean. (2) Every companion matrix in Mn(R) is strongly clean. (3) R is an n-SRC ring. (4) For every x0, x1 ? R with x0 ? x1 2U(R), every monic polynomial h ? R[t] of degree n has an SRC factorization relative to (x0, x1). (5) For every x0, · · · , xk ? R with xi ? xj 2U(R) whenever i 6= j, every monic polynomial h ? R[t] of degree n has an SRC factorization relative to (x0, x1, · · · , xk). And finally we give some 14.4pt; mso-yfti-irow: 12" clean. for example, we proof following corollary. Corollary 41.4: Let P be a prime number, G a finite abelian p?group, and let R be a commu-tative local ring for which p = 0 in R. Then, RG is a n ? SRC if and only if R is n ? SRC. Furthermore, our results will provide a large 14.5pt; mso-yfti-irow: 14" are not strongly clean, giving further negative answers to [25, Question 5]. Also we proof following corollary. Corollary 50:4: Let (R, J) be a commutative local ring. Then, the following are equivalent: (1) M2(R) is strongly clean. (2) R is a 2 ? SRC ring. (3) The polynomial t2 ? t + j ? R[t] has a root for every j ? J. If Char(R/J) 6= 2(equivalently, 2 2U(R)), the above are equivalent to (4) Every element of the multiplicative group 1 + J has a square root (in 1 + J). (5) Every element of 1 + J has a square root in R.
حلقه یکدار R را قویاً تمیز گوییم اگر هر عنصر آن را بتوان به صورت مجموع یک یکه و یک خودتوان نوشت که جا به جا می شوند. در این پایان نامه به مشخصه سازی حلقه های موضعی تعویض پذیرR که (R)nM قویاً‌تمیز است با توجه به فاکتورگیری هایی در [t]R می پردازیم. هم چنین ثابت می کنیم که برای هر چندجمله ای تکین چون[t]R?f ، قویاً تمیزی ماتریس همراه f با قویاً‌تمیزی تمام ماتریس با چندجمله ای مشخصه f معادل است . و سرانجام به معرفی کلاس های خاصی از سازی حلقه های موضعی تعویض پذیرR که (R)nM قویاً‌تمیز است می پردازیم .

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