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الكلية كلية التربية للعلوم الصرفة     القسم  قسم الرياضيات     المرحلة 7
أستاذ المادة اسعد محمد علي حسين الحسيني       25/12/2018 17:06:29
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 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.
2.2.1 Definition. Let M be a right R-module. A subset A of M is called a submodule of M, notationally A M (or also AR MR) if A is a right R -module with respect to the restriction of the addition and module multiplication of M to A.
2.2.2 LEMMA. Let M be a right R-module. If A is a subset of M and A? ? then the following are equivalent:
(1) A M.
(2) A is a subgroup of the additive group of M and for all a A and all r R we have ar A (where ar is the module multiplication in M).
(3) For all a1, a2 A, a1 + a2 A (with respect to addition in M) and for all a A and all r R, we have ar A.

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