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68 Cards in this Set
- Front
- Back
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A ring R is...
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...a set that with two binary operations (multiplication, addition) s.t. it is an abelian group under addition and associative multiplication
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A unity in a ring is...
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...a non-zero element that is an identity under multiplication
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A unit of a ring is...
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...a non-zero element that has a multiplicative inverse
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What are the first 4 properties of elements of a ring? (Let a,b,c <R)
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(1) a0=0a=0
(2) a(-b)=-a(b)=-(ab) (3) (-a)(-b)=ab (4) a(b-c)= ab-ac and (b-c)a= ba-ca |
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What are the multiplicative properties of a ring with unity element 1?
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(1) -1(a)=-a
(2) (-1)(-1)=1 |
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If a ring has a unity...
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...it is unique
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If a ring element has a multiplicative inverse...
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...it is unique
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If a,b and c (and a does not equal to 0) belong to a ring and ab=ac then...
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...we can't conclude that b=c
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A subset S of a ring R is a subring if...
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...S is itself a ring with the operations of R
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A nonempty subset S of a ring R is a subring if...
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...S is closed under subtraction and multiplication. Or a-b and ab are in S whenever a and b are
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A zero divisor is...
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...a nonzero element a of a commutative ring R s.t. there is a nonzero element b contained in R with ab=0
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An integral domain...
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...is a commutative ring with unity and no zero divisors
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Let a,b,c belong to an integral domain. If a is nonzero and ab=ac then...
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...b=c
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A finite integral domain...
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...is a field
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For every prime p, Zp...
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...is a field
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The characteristic of a ring R is...
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...the least positive integer n s.t. nx=0 for all x in R. If no such integer exists, we say that R has characteristic 0. The characteristic of R is denoted char R
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Let R be a ring with unity 1. If 1 has infinite order under addition...
If 1 has order n under addition... |
...then char R=0
...then char R=n |
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The characteristic of an integral domain is...
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...zero or prime
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What are some examples of rings that are integral domains?
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Ring of integers, ring of integers modulo p when p is prime, ring of polynomials with integer coefficients
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What are some examples of rings that are not integral domains?
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Ring of 2 by 2 matrices over the integers, the ring of integers modulo n when n is not prime, the external product of the integers (Z+Z)
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A field is...
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...a commutative ring with unity in which every nonzero element is a unit
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A subring A of a ring is called an ideal if...
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...for every r in R and every a in A, ra and ar is contained in A
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A nonempty subset A of a ring R is called an ideal if...
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(1) a-b is in A, whenever a and b are
(2) ra and ar are in A whenever a is in A and r is in R |
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Let R be a ring and let A be a subring of R. The set of cosets is a ring under....
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...the operation of coset addition and multiplication IFF A is an ideal
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An ideal A of R is called a proper ideal if...
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...A is a proper subset of R
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A prime ideal A of a commutative ring R is...
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...a proper ideal of R such that a,b in R and ab in A implies either a or b in A
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A maximal ideal of a commutative ring R is...
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...a proper ideal such that whenever B is an ideal of R and A is contained in B which is contained in R, either A=B or B=R
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Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is an integral domain IFF...
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...A is prime
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Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is a field IFF...
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...A is maximal
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A ring homomorphism T from a ring R to a ring S is...
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...a mapping from R to S that preserves the two ring operations
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When is a ring homomorphism an isomorphism?
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When the mapping is both one to one and onto
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What is the natural homomorphism from Z to Zn?
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A mapping k-> k mod n
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What are some examples of ring homomorphisms?
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Nat. homomorphism from Z to Zn, mapping from a+bi--> a-bi (isomorphism), f(x)--> f(1)
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Let R be a commutative ring with characteristic 2. Then the mapping from a--> a^2...
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...is a ring homomorphism from R onto R
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What are the first 4 properties of ring homomorphisms? (Let T be a mapping from R onto S. Let A be a subring of R, let B be an ideal of S)
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(1) For any r<R, and any positive integer n, T(nr)=n*T(r) and T(r^n)=(T(r))^n
(2) T(A) is a subring of S (3) If A is an ideal and T is onto, T(A) is an ideal (4) T^(-1)(B) is an ideal of R |
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What are the last 4 properties of ring homomorphisms? (Let T be a mapping from R onto S. Let A be a subring of R, let B be an ideal of S)
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(5) If R is commutative, then T(R) is commutative
(6) If R has unity 1, S is not the zero element, and T is onto, then T(1) is the unity of S (7) T is an isomorphism IFF T is onto and Ker T={0} (8) If T is an isomorphism from R onto S, then T^(-1) is an isomorphism from S onto R |
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Let T be a ring homomorphism from R to a ring S. Then Ker T...
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...is an ideal of R
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What is the first isomorphism theorem for rings?
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Let T be a ring homomorphism from R to S. Then the mapping from R/Ker T to T(R), given by r+Ker T --> T(r), is an isomorphism
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Every ideal of a ring is...
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...the kernel of a ring homomorphism of R
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Let R be a ring with unity 1. The mapping T: Z-->R...
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... given by n-->n*1 is a ring homomorphism
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If R is a ring with unity and the characteristic of R is n>0, then R...
If the characteristic of R is 0, then R... |
...contains a subring isomorphic to Zn
...contains a subring isomorphic to Z |
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If F is a field of characteristic p, then F...
If F is a field of characteristic 0, then F... |
...contains a subfield isomorphic to Zp
...contains a subfield isomorphic to the rational numbers |
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Let D be an integral domain. Then there exists a field F (called the field of quotients)...
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...that contains a subring isomorphic to D
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If D is an integral domain then D[x]...
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...is an integral domain
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Let F be a field and let g(x), f(x) be contained in F[x] with g(x) nonzero. Then...
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...there exist unique polynomials q(x) and r(x) in F[x] such that f(x)= g(x)q(x)+r(x) and either r(x)=0 or deg r(x) < deg g(x)
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What does the remainder theorem say?
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Let F be a field and let a{F, and f(x) { F[x]. Then f(a) is the remainder in the division of f(x) by x-a
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What does the factor theorem say?
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Let F be a field and let a{F, and f(x) { F[x]. Then a is a zero of f(x) IFF x-a is a factor f(x)
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A polynomial of degree n over a field has at most...
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... n zeros, counting multiplicities
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A principal ideal domain is...
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... an integral domain R in which every ideal has the form <a>={ra | r in R} for some a in R
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Let F be a field. Then F[x] is...
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...a principal ideal domain
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Let F be a field, T a nonzero ideal in F[x], and g(x) an element of F[x]. Then T= <g(x)> IFF...
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...g(x) is a nonzero polynomial of minimum degree in T
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Let D be an integral domain. A polynomial f(x) from D[x] (that is neither a unit nor the zero polynomial) is said to be irreducible over D...
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...if whenever f(x)=g(x)h(x) with g(x), h(x) in D[x], then g or h is a unit in D[x]
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Let F be a field. If f(x) is in F[x] and deg f(x) is 2 or 3 then...
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...f(x) is reducible over F if and only if f(x) has a zero in F
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Let f(x) be in Z[x]. If f(x) is reducible over...
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...Q, then it is reducible over Z
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What are the key details about the mod p test?
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p must be a prime, f(x) must be in Z[x], etc.
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Let F be a field. Then <p(x)> is a maximal ideal IFF...
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...p(x) is irreducible over F
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Let F be a field and let p,a,b be in F[x]. If p is irreducible over F and p(x) | a(x)b(x) then...
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...p(x) divides a(x) or b(x)
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Elements a,b of an integral Domain are called associates...
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...if a=ub means that u is a unit
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A nonzero element of an integral domain D is called an irreducible if a is not a unit and whenever b,c are in D with a=bc, then...
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...b or c is a unit
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A nonzero element a of an integral domain D is called a prime if a is not a unit and a | bc implies...
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...a|b or a|c
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In an integral domain, a irreducible...
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...is not necessarily a prime. But every prime is an irreducible
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In a PID, an element is prime IFF...
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...it is an irreducible
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An integral domain is a UFD if...
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...(1) Every nonzero element of D that is not a unit can be written as a product of irreducibles, and
(2) the factorization into irreducibles is unique up to associates and the order in which the factors appear |
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In a PID, any strictly increasing chain of ideals...
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...must be finite in length
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Let F be a field. F[x] is...
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...a PID and a UFD
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An integral domain D is called a Euclidean Domain if...
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...there is a function d ('measure') from the nonzero elements of D to the nonnegative integers s.t.
(1) d(a) < d(ab) for all a,b in D (2) if a,b in D, b is nonzero, then there exist q,r in D s.t. a=bq+r, where r=0 or d(r) < d(b) |
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Euclidean Domain implies...
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...PID and that implies UFD
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If D is a UFD then D[x]...
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...is a UFD
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