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44 Cards in this Set
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GCD

If a≠0 or b≠0, d=gcd(a,b) if:
1 da and db 2 if ca and cb, then c≤d. 
Prime

a is prime if
1 a≠±1,0 2 the only divisors of a are ±1 and ±a 
a≡b (mod n)

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

Equivalence Relation

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. 
Congruence class of a (mod n)

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

Z_n

The set of all congruence classes modulo n

Ring

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 
Commutative ring

a ring with the added axiom that multiplication is commutative

Ring with identity

a ring that contains a multiplicative identity 1

M(R)

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

Rings of functions

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

Integral domain

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.

Field

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.

Subring, subfield

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.

Cartesian product

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 componentwise; A X B is commutative if both A and B are and A X B has identity if both A and B do.

Unit

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.

Associate

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.

Multiplicative Inverse

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

Zero divisor

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. 
Ring isomorphism

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

Ring homomorphism

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

Image of a function f

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

R[x]

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

GCD of polynomials f(x) and g(x)

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

Irreducible polynomial

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

Polynomial function

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

Root of a polynomial

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))

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

F[x]/p(x)

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

Extension Field

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.

Ideal

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.

Principal ideal

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)

a ≡ b (mod I) if ab is in I

Left coset a+I

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

Quotient ring

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

Homomorphic image

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

Prime ideal

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

Maximal ideal

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.

Field of quotients of R

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

Group

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 
GL(2,K)

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

Center of a group

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

Cyclic subgroup <a> of G generated by a

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.

Cyclic group

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