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A #~{module} over a ring ~R, or ~R-#~{module}, is an (additive) abelian group ~M , with external operation _ ~R # ~M -> ~M _ where _ ( ~r , ~x ) -> ~r ~x , _ such that, for _ &lamda. , &mu. &in. ~R , ~x , ~y &in. ~M :

- ( λ + &mu. ) ~x _ = _ λ ~x + &mu. ~x
- ( λ &mu. ) ~x _ = _ λ ( &mu. ~x )
- λ ( ~x + ~y ) _ = _ λ ~x + λ ~y

[ Note that a right ideal, ~I, of ~R is an ~R-module since _ ~r ~i &in. ~I , _ &forall. ~r &in. ~R , ~i &in. ~I ]

We have: _ 0_~R ~x = 0_~M , _ &forall. ~x &in. ~M. _ [ we will drop the suffixes when it is clear which zero element is being referred to. ]

For if _ 0 ~x = ~y != ~0 , _ then _ λ ~x = ( λ + 0 ) ~x = λ ~x + ~y != λ ~x _ contradiction.

If ~R has a identity element 1_~R , and _ 1_~R ~x _ = _ ~x , &forall. ~x &in. ~M, then ~M is called a #~{unital} ~r-module.

A module over a field ~F is called a ~#{linear space} or ~#{vector space} over ~F

An #~{algebra} over a field ~F is a linear space over ~F , which has a multiplicative operation such that it is itself a ring.

We are all familiar with polynomials with real valued "coefficients", ~a_0 , ~a_1 , ... , ~a_~n , _ and a "variable", ~t , such that a typical polynomial is written

- ~f ( ~t ) _ = _ ~a_0 + ~a_1 ~t + ~a_2 ~t ^2 + ... + ~a_~n ~t ^~n

We will now formalize the notion of a polynomial.

If ~R is a commutative ring, then a #~{polynomial}, ~f over ~R is a sequence: _ ~f _ = _ ( ~a_0 , ~a_1 , ... ) _ = _ ( ~a_~i )_~i , _ of elements of ~R, where only a finite number of the elements are non-zero. The elements, _ ~a_~i , _ are called the #~{coefficients} of ~f.

If not all the ~a_~i are zero, the #~{degree} of ~f , written d ( ~f ) , is the largest integer, ~n , for which _ ~a_~n _ is non-zero. If all the ~a_~i are zero, then we write _ d ( ~f ) = -&infty. . _ [ &forall. ~n &in. #Z , _ -&infty. =< ~n , _ and _ -&infty. + ~n = -&infty. ]

Let _ ~f = ( a_~i )_~i _ and _ ~g = ( ~b_~i )_~i _ be polynomials over a ring ~R. Define their

- #~{sum}: _ ~f + ~g _ = _ ( ~a_~i )_~i + ( ~b_~i )_~i = ( ~p_~i )_~i _ where _ ~p_~i = ~a_~i + ~b_~i .
- #~{product}: _ ~f ~g _ = _ ( ~a_~j )_~j ( ~b_~k )_~k = ( ~q_~i )_~i _ where _ ~q_~i = &sum._{~j+~k=~i} ~a_~j ~b_~k .

We have: _ d ( ~f + ~g ) = max \{ d ( ~f ) , d ( ~g ) \} , _ and _ d ( ~f ~g ) =< d ( ~f ) + d ( ~g ) _ [ not always equality as ~R is not necessarily an integral domain and we could have _ ~a_~j ~b_~k = 0 , _ ~a_~j != 0 , ~b_~k != 0 .]

With the operations defined above, the set, ~S, of all polynomials over ~R is itself a ring.

That ~S is additively associative is fairly obvious.

The zero element is the polynomial with all zero coefficients, ( 0, 0, ... ), and _ -~f _ = _ ( -~a_~i )_~i _ = _ ( -~a_0 , -~a_1 , ... ). As ~R is additively commutative, ~S is an abelian group with respect to addition.

~S is multiplicatively associative: _ let _ ~f = ( ~a_~i )_~i , ~g = ( ~b_~j )_~j , ~h = ( ~c_~k )_~k &in. ~S,

( ~f ~g ) ~h _ = _ ( &sum._{~l+~k=~m} ( &sum._{~i+~j=~l} ~a_~i ~b_~j ) ~c_~k )_~m _ = _ ( &sum._{~i+~j+~k=~m} ~a_~i ~b_~j ~c_~k )_~m _ = _ ~f ( ~g ~h ) , _ by symmetry.

The operations are distributive

( ~f + ~g ) ~h _ = _ ( ~a_~i + ~b_~i )_~i ( ~c_~j )_~j _ = _ ( &sum._{~i+~j=~k} ( ~a_~i + ~b_~i ) ~c_~j )_~k _ = _ ( &sum._{~i+~j=~k} ( ~a_~i ~c_~j ) )_~k + ( &sum._{~i+~j=~k} ( ~b_~i ~c_~j ) )_~k _ = _ ~f ~h + ~g ~h

If ~R has identity element 1_~R then 1_~S = ( 1_~R, 0, 0, ... ) is the identity element in ~S. _ 1_~S ~f = ( &sum._{~i+~j=~k} ( ~I_i ~a_~j ) )_~k