Read More
Date: 5-8-2021
1537
Date: 10-8-2021
1925
Date: 25-6-2017
1211
|
Definition Let R be a unital ring. A set M is said to be a left module over the ring R (or left R-module) if (i) given any x, y ∈ M and r ∈ R, there are well-defined elements x + y and rx of M,
(ii) M is an Abelian group with respect to the operation + of addition,
(iii) the identities
r(x + y) = rx + ry, (r + s)x = rx + sx,
(rs)x = r(sx), 1Rx = x
are satisfied for all x, y ∈ M and r, s ∈ R, where 1R denotes the multiplicative identity element of the ring R.
Definition Let R be a unital ring. A set M is said to be a right module over R (or right R-module) if (i) given any x, y ∈ M and r ∈ R, there are well-defined elements x + y and xr of M,
(ii) M is an Abelian group with respect to the operation + of addition,
(iii) the identities
(x + y)r = xr + yr, x(r + s) = xr + xs,
x(rs) = (xr)s, x1R = x
are satisfied for all x, y ∈ M and r, s ∈ R, where 1R denotes the multiplicative identity element of the ring R.
If the unital ring R is a commutative ring then there is no essential distinction between left R-modules and right R-modules. Indeed any left module M over a unital commutative ring R may be regarded as a right module on defining xr = rx for all x ∈ M and r ∈ R. We define a module over a unital commutative ring R to be a left module over R.
Example If K is a field, then a K-module is by definition a vector spaceover K.
Example Let (M, +) be an Abelian group, and let x ∈ M. If n is a positive integer then we define nx to be the sum x + x + · · · + x of n copies of x. If n is a negative integer then we define nx = −(|n|x), and we define 0x = 0.
This enables us to regard any Abelian group as a module over the ring Z of integers. Conversely, any module over Z is also an Abelian group.
Example Any unital commutative ring can be regarded as a module over itself in the obvious fashion.
Let R be a unital ring that is not necessarily commutative, and let + and × denote the operations of addition and multiplication defined on R. We denote by Rop the ring (R, +, ×-), where the underlying set of Rop is R itself, the operation of addition on Rop coincides with that on R, but where the operation of multiplication in the ring Rop is the operation ×- defined so that r×-s = s × r for all r, s ∈ R. Note that the multiplication operation on the ring Rop coincides with that on the ring R if and only if the ring R is commutative.
Any right module over the ring R may be regarded as a left module over the ring Rop. Indeed let MR be a right R-module, and let r.x = xr for all x ∈ MR and r ∈ R. Then
r.(s.x) = (s.x)r = x(sr) = x(r×-s) = (r×-s).x
for all x ∈ MR and r, s ∈ R. Also all other properties required of left modules over the ring Rop are easily seen to be satisfied. It follows that any general results concerning left modules over unital rings yield corresponding results concerning right modules over unital rings.
Let R be a unital ring, and let M be a left R-module. A subset L of M is said to be a submodule of M if x + y ∈ L and rx ∈ L for all x, y ∈ L and r ∈ R. If M is a left R-module and L is a submodule of M then the quotient group M/L can itself be regarded as a left R-module, where r(L+x) ≡ L+rx for all L + x ∈ M/L and r ∈ R. The R-module M/L is referred to as the quotient of the module M by the submodule L.
A subset L of a ring R is said to be a left ideal of R if 0 ∈ L, −x ∈ L, x + y ∈ L and rx ∈ L for all x, y ∈ L and r ∈ R. Any unital ring R may be regarded as a left R-module, where multiplication on the left by elements of R is defined in the obvious fashion using the multiplication operation on the ring R itself. A subset of R is then a submodule of R (when R is regarded as a left module over itself) if and only if this subset is a left ideal of R.
Let M and N be left modules over some unital ring R. A function ϕ: M → N is said to be a homomorphism of left R-modules if ϕ(x + y) = ϕ(x) + ϕ(y) and ϕ(rx) = rϕ(x) for all x, y ∈ M and r ∈ R. A homomorphism of Rmodules is said to be an isomorphism if it is invertible. The kernel ker ϕ and image ϕ(M) of any homomorphism ϕ: M → N are themselves R-modules.
Moreover if ϕ: M → N is a homomorphism of R-modules, and if L is a submodule of M satisfying L ⊂ ker ϕ, then ϕ induces a homomorphism ϕ-: M/L → N. This induced homomorphism is an isomorphism if and only if
L = ker ϕ and N = ϕ(M).
Definition Let M1, M2, . . . , Mk be left modules over a unital ring R. The direct sum M1⊕M2⊕· · ·⊕Mk of the modules M1, M2, . . . , Mk is defined to be the set of ordered k-tuples (x1, x2, . . . , xk), where xi ∈ Mi for i = 1, 2, . . . , k.
This direct sum is itself a left R-module, where
(x1, x2, . . . , xk) + (y1, y2, . . . , yk) = (x1 + y1, x2 + y2, . . . , xk + yk),
r(x1, x2, . . . , xk) = (rx1, rx2, . . . , rxk)
for all xi, yi ∈ Mi and r ∈ R.
If K is any field, then Kn
is the direct sum of n copies of K.
Definition Let M be a left module over some unital ring R. Given any subset X of M, the submodule of M generated by the set X is defined to be the intersection of all submodules of M that contain the set X. It is therefore the smallest submodule of M that contains the set X. A left R-module M is said to be finitely-generated if it is generated by some finite subset of itself.
Lemma 1.4 Let M be a left module over some unital ring R. Then the submodule of M generated by some finite subset {x1, x2, . . . , xk} of M consists of all elements of M that are of the form
r1x1 + r2x2 + · · · + rkxk
for some r1, r2, . . . , rk ∈ R.
Proof The subset of M consisting of all elements of M of this form is clearly a submodule of M. Moreover it is contained in every submodule of M that contains the set {x1, x2, . . . , xk}. The result follows.
|
|
مخاطر عدم علاج ارتفاع ضغط الدم
|
|
|
|
|
اختراق جديد في علاج سرطان البروستات العدواني
|
|
|
|
|
نشر مظاهر الفرح في مرقد أبي الفضل العباس بذكرى ولادة السيدة زينب (عليهما السلام)
|
|
|