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موديولات/ماجستير/كورس2/محاضرة6

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الكلية كلية التربية للعلوم الصرفة     القسم  قسم الرياضيات     المرحلة 7
أستاذ المادة اسعد محمد علي حسين الحسيني       26/02/2019 16:46:36
1. Modules
Each ring R which we consider will be assumed to contain a multiplicative
identity element, denoted by 1; which differs from its additive identity element
0. Now let R be a ring.
Definition: A left R-module consists of an additive abelian group A together
with a function ? : R × A ? A which satisfies the following four conditions:
1. ? (r, a + a ) = ? (r, a) + ? (r, a );
2. ? (r + r , a) = ? (r, a) + ? (r , a);
3. ? (rr , a) = ? (r, ? (r , a)); and
4. ? (1, a) = a
for all elements r and r of R and all elements a and a of A.
By convention, the element ? (r, a) is denoted by r •a or by ra; and ? is said to provide an operation of scalar multiplication on the abelian group A. Using this juxtaposition notation instead of the function ?, the above four conditions become:
1. r(a + a ) = ra + ra ;
2. (r + r )a = ra + r a;
3. (rr )a = r(r a) and
4. 1a = a

Examples:
1. If R is a field, then left R-modules are also called vector spaces over R.
2. Every additive abelian group A may be regarded as a left Z-module, where Z is the ring of integers, in precisely one way: for n ? Z and a ? A, define na as follows:
1. If n > 0 then na is the sum of n copies of a;
2. If n = 0 then na = 0;
3. If n < 0; then na is the sum of -n copies of –a.
3. Any ring R can be considered as a left R-module in the following manner:
for the abelian group, use R with its additive structure. For the scalar multiplication
of the ring R on the abelian group (R, +), use the ring multiplication.
4. The "smallest" of all left R-modules is the one having precisely one element,
namely an additive identity. We will consistently denote this module by (0).


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