Abstract Algebra Final (MATH 371)

If a≠0 or b≠0, d=gcd(a,b) if:
1- d|a and d|b
2- if c|a and c|b, then c≤d.

a is prime if
1- a≠±1,0
2- the only divisors of a are ±1 and ±a

a≡b (mod n) if a,b, and n>0 are integers and a-b is a multiple of n.

An equivalence relation R on A is a subset of A X A that is:
1- reflexive: (a,a) is in R for all a in A
2- symmetric: if (a,b) is in R, (b,a) is in R
3- transitive: if (a,b) and (b,c) are in R, then (a,c) is in R.

The set of all integers b such that b≡a (mod n).

The set of all congruence classes modulo n

A NONEMPTY set equipped with two operations, usually called addition and multiplication, with eight axioms:
1- Closure under addition
2- Closure under multiplication
3- Associativity under addition
4- Associativity under multiplication
5- Commutativity under addition
6- The existence of the additive identity 0
7- The existence of the additive inverse -a of each element a
8- Distributivity on the right and on the left

a ring with the added axiom that multiplication is commutative

a ring that contains a multiplicative identity 1

The set of all 2X2 matrices over the real numbers; this is a noncommutative ring with identity.

Rings whose elements are functions over a set. The operations are defined as function addition and function multiplication.

a COMMUTATIVE ring R WITH IDENTITY 1≠0 is an integral domain if whenever a,b are in R and ab=0, either a=0 or b=0.

a COMMUTATIVE ring R WITH IDENTITY 1≠0 is a field if for each a≠0 in R, the equation ax=1 has a solution in R.

A subring is a subset of a ring that satisfies all of the ring axioms under the same operations as defined for the parent ring. The same goes for a subfield, replacing the word ring with field.

The cartesian product of two sets A and B is the set A X B of all ordered pairs (a,b), where a is in A and b is in B. If A and B are rings, A X B is a ring where operations are defined component-wise; A X B is commutative if both A and B are and A X B has identity if both A and B do.

An element a of a ring R WITH IDENTITY is a unit if there exists some element b in R such that ab=1=ba. b is the multiplicative inverse.

An element a of a commutative ring R is an associate of an element b in R if there exists a unit u in R such that a=bu.

The multiplicative inverse of the nonzero element a in a ring R WITH IDENTITY is the nonzero element b for which ab=1=ba.

An element a in a ring R is a zero divisor if:
1- a≠0
2- There exists some b≠0 in R such that ab=0.

A bijective function f between rings that has these properties: f(a+b)=f(a)+f(b)

A function f between rings that has these properties: f(a+b)=f(a)+f(b)

The set of all elements of the codomain to which f maps an element of the domain.

The ring R with an indeterminate element x not in R included. In other terms, the set of all polynomials with coefficients in R.

The monic polynomial of highest degree that divides both f and g

A nonconstant polynomial in F[x], where F is a FIELD, whose only divisors are its associates and the constant polynomials (units)

A function on a ring R induced by a polynomial in R[x]

A root of a polynomial p(x) in R[x] is an element of R that is mapped to 0 by the polynomial function induced by p(x).

f(x) ≡ g(x) (mod p(x)) if there exists a polynomial q(x) such that f(x) -g(x)=q(x)p(x).

The set of all congruence classes modulo p(x) in F[x].

A field created by including new elements in a previously defined field. The ring F[x]/p(x) may be an extension field of F.

A subring I of a ring R with the following property: if a is in I and r is in R, then ar and ra are both in I.

An ideal in a COMMUTATIVE ring WITH IDENTITY generated by a single element. In other words, the set of all multiples of an element of the ring.

a ≡ b (mod I) if a-b is in I

The set of all elements b of a ring such that a-b is in I.

The ring of all cosets modulo I, where is an ideal.

S is a homomorphic image of R if f:R→S is a surjective homomorphism of rings

An ideal I in a COMMUTATIVE ring R that satisfies the following property: if a and b are in R and ab is in I, then either a is in I or b is in I

An ideal I of the COMMUTATIVE ring R that satisfies the following: if M is an ideal in R such that I is a subring of M, then I=M or M=R.

The field constructed by creating quotients of the elements of the INTEGRAL DOMAIN R.

A NONEMPTY set equipped with a binary operation * that satisfies the following axioms:
1- closure under *
2- associativity of *
3- existence of the identity
4- existence of the inverse of every element

The multiplicative group of units in the ring of 2X2 matrices with entries in K.

The set of all elements in a group that commute with every element of the group.

A subgroup of G generated by the element a of G is a subgroup whose elements are all of the powers of a. This subgroup <a> is abelian.

A group that is entirely generated by one element. This group is abelian.