موديولت 3 دكتوراه

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.