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الكلية كلية التربية للعلوم الصرفة     القسم  قسم الرياضيات     المرحلة 7
أستاذ المادة اسعد محمد علي حسين الحسيني       06/10/2018 19:31:11
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.

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.

Examples and remarks
(1) A ring R can be considered as a right R-module RR, as a left -module RR and as an R-R bimodule RRR respectively. A right ideal, left ideal or two-sided ideal of R is then a submodule of RR, of RR or of RRR respectively.
(2) Every module M possesses the trivial submodules 0 and M, where 0 is the submodule which contains only the zero element of M.
(3) Let M be arbitrary and let m0 ? M. Then, as we see immediately from 2.2.2,
m0R = { m0r| r ? R } is a submodule of M which is called the cyclic submodule of M generated by m0.
(4) If MK is a vector space over the field K then the submodules are called (linear) subspaces.
(5) In the ring Z of integer numbers every ideal is cyclic.
(6) Cyclic ideals of a ring are called principal ideals and a commutative ring is called a principal ideal ring if every ideal is a principal ideal.
(7) A field K has only the trivial ideals 0 and K.

2.2.3 Definition
( 1 ) A module M =MR is called cyclic : <=> ?m0 ? M [M = m0R ]
(2) A module M =MR is called simple : <=>M ? 0 ? ? A ? M [ A = 0 ? A = M ] ,
i.e. M ? 0 and 0 and M are the only submodules of M.
(3) A ring R is called simple : <=> R? 0? ? A ? RRR [ A = 0 ? A = R ],
i.e. R? 0and 0 and R are the only two-sided ideals of R.
(4) A submodule A? M is called a minimal, respectively a maximal, submodule of M : <=> 0?A and ? B ? M[ B?(?)A?B=0] , respectively
A?(?)M ?? B ? M[ A?(?)B?B=M]

Note: Minimal submodules are previously the simple submodules. The minimal (=simple), respectively maximal, submodules of a module are, if they exist, evidently minimal, respectively maximal, elements in the ordered set of non-zero, respectively proper, submodules under the ordering by inclusion.

2.2.4 LEMMA. M is simple ?[M ? 0 ??m ? M [m ? 0 ? mR = M].

Examples
(1) Z contains no minimal (=simple) ideal, for if n Z ? 0 then, for example, 2n Z is a non-zero ideal properly contained within n Z. The maximal ideals of Z are exactly the prime ideals p Z, p =prime number.
(2) Q_Z has no minimal and no maximal submodules.
(3) In a vector Space V = VK the minimal (=simple) subspaces are just the one-dimensional subspaces and these are given precisely in the form vK by the elements v ? V, v ? 0. If V is n-dimensional, then the maximal subspaces are exactly the (n - l)-dimensional subspaces. If V is not finite dimensional, then there are likewise maximal subspaces (a fact which is well known from linear algebra and which will here be shown later).
(4) If K is a skew field, then KK is simple as also is K as a ring (i.e. KKK is simple).
This follows immediately from the fact that every element ?0 of K possesses an inverse.
(5) Let R := Kn be the ring of n × n square matrices with coefficients in a skew field. Without proof we mention (proof follows later) that although R is simple (as a ring) nevertheless RR is not for n > 1.

2.3.1 LEMMA. Let ? be a set of submodules of a module M, then
?_A???A:= {m ? M | ? A ?? [m ? A]} is a submodule of M.

Remark. We note that when ? =? this definition yields ?_A???A:=M

From 2.3.1 there follows immediately the corollary.

COROLLARY. ?_A???A is the biggest submodule of M which is contained in all A ??.

Examples 2Z?3Z = 6Z, ?_(pis prime)?? p? Z= 0.


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