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الكلية كلية التربية للعلوم الصرفة
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المرحلة 7
أستاذ المادة اسعد محمد علي حسين الحسيني
07/10/2018 06:00:05
In the following all rings, which are mostly denoted by R, S or T, are to possess a unit element 1.
2.1.1 Definition. Let R be a ring. A right R-module M is (I) an additive abelian group M together with (II) a mapping M ×R? M with (m, r) ? mr, called module multiplication, for which we have (1) Associative law: (mr1)r2 = m(r1r2). (2) Distributive laws: (m1 + m2)r = m1r + m2r, m(r1 + r2) = m r1 + mr2. (3) Unitary law: m1= m. (In the above m, m1, m2 are arbitrary elements from M and r, r1, r2 are arbitrary elements from R).
Notes: 1. According to this definition all modules in the following are unitary. 2. If M is a right R-module, then we write also MR or M = MR in order to indicate the ring which is involved. 3. An analogous definition holds for left modules. 4. If S and R are two rings then M is an S-R-bimodule if M is a left S-module and a right R-module (with the same additive abelian group) and if, additionally, the following associative law holds: s (mr) = (sm)r for arbitrary s ? S, m ? M , r ? R. We write also SMR for the S-R-bimodule. 5. It is well known that an R-module is called a linear vector space over R if R is a field (or skew field). 6. The modules over the ring Z of integer numbers are the abelian groups (written additively). 7. If M is a right R -module we denote the neutral element of the additive group of M by 0M and that of the additive group of R by 0R , then it follows that 0M r= 0M, m0R = 0M, 8. And also -(mr) = (-m)r = m(-r) for arbitrary m ? M, r ? R. 9. In the following we write 0, as is usual, both for 0M and for 0R.
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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