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),                                                              1_Rx = x

are satisfied for all x, y ∈ M and r, s ∈ R, where 1_R 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,                                 x1_R = x

are satisfied for all x, y ∈ M and r, s ∈ R, where 1_R 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 ∈ M_R 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 R^op 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⊕· · ·⊕M_k of the modules M₁, M₂, . . . , M_k is defined to be the set of ordered k-tuples (x₁, x₂, . . . , x_k), where x_i ∈ M_i  for i = 1, 2, . . . , k.

This direct sum is itself a left R-module, where

                          (x₁, x₂, . . . , x_k) + (y₁, y₂, . . . , y_k) = (x₁ + y₁, x₂ + y₂, . . . , x_k + y_k),

                                              r(x₁, x₂, . . . , x_k) = (rx₁, rx₂, . . . , rx_k)

for all x_i, y_i ∈ M_i 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, . . . , x_k} of M consists of all elements of M that are of the form

                          r₁x₁ + r₂x₂ + · · · + r_kx_k

for some r₁, r₂, . . . , r_k ∈ R.