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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- d|a and d|b 2- if c|a and c|b, 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 a-b 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 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. 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 a-b is in I Left coset a+I The set of all elements b of a ring such that a-b 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 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 is abelian. Cyclic group A group that is entirely generated by one element. This group is abelian.