Abstract Algebra

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This page helps students who have encoutered terms from abstract algebra and need to quickly brush up their knowledge. For in depth discussion of abstract algebra, go to In Depth WikiBook on Abstract algebra.

Definitions

• Group

A group (G,.) is a nonempty set G together with a binary operation . on G such that the following conditions hold:
(i) Closure: For all a,bG the element a.b is a uniquely defined element of G.
(ii) Associativity: For all a,b,cG, we have a.(b.c) = (a.b).c.
(iii) Identity: There exists an identity element eG such that e.a=a and a.e=a for all aG.
(iv) Inverses : For each aG there exists an inverse element a-1G such that a.a-1=e and a-1.a=e.

We will usually simply write ab for the product a.b.

• Ring

Let R be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and .. Then R is called a commutative ring with respect to these operations if the following properties hold:
(i) Closure: If a,bR, then the sum a+b and the product a.b are uniquely defined and belong to R.
(ii) Associative laws: For all a,b,cR,
a+(b+c) = (a+b)+c and a.(b.c) = (a.b).c.
(iii) Commutative laws: For all a,bR,
a+b = b+a and a.b = b.a.
(iv) Distributive laws: For all a,b,cR,
a.(b+c) = a.b + a.c and (a+b).c = a.c + b.c.
(v) Additive identity: The set R contains an additive identity element, denoted by 0, such that for all aR,
a+0 = a and 0+a = a.
(vi) Additive inverses: For each aR, the equations
a+x = 0 and x+a = 0
have a solution xR, called the additive inverse of a, and denoted by -a.
The commutative ring R is called a commutative ring with identity if it contains an element 1, assumed to be different from 0, such that for all aR,
a.1 = a and 1.a = a.
In this case, 1 is called a multiplicative identity element or, more generally, simply an identity element.

As with groups, we will use juxtaposition to indicate multiplication, so that we will write ab instead of a.b.

• Field

A field is a
ring, such that for any a that is not equal to 0, there is an element b that is inverse to a with respect to multiplication: ab=1.

Examples

• A set of remainders by modulo M, with respect to addition, is a group.
• A set of integer numbers is a ring, with respect to addition and multiplication.
• A set of real numbers (as well as the set of complex numbers) is a field.
• A set of remainders by modulo M, with respect to addition and multiplication (both by modulo M), is a field if M is prime.
• The set of all rotation in Euclidean spaces is a group.
• The set of all matrices is a ring.
• Any field is a ring and any ring is a group.