Flashcards
ALGEBRA DEFINITIONS
Flashcards do conjunto (35)
- 1RINGA ring is a set R with two binary associative operation + and X s.t. :
(1) Under addition, R forms an abelian group
(2) R is closed under multiplication
(3) Multiplication is commutative
(4) Multiplication distributes over addition
(5) R has a multiplicative identity - 2UNITWith R a ring and uϵR, we say u is a unit if there is some vϵR s.t. uv=1.
- 3SUBRINGLet R≠{0} be a ring. A Subset S of R is a subring if S≠{0} and S is itself a ring with 1(S)=1(R).
- 4HOMOMORPHISMLet R, R' be rings. A map φ:R->R' is a ring homomorphism if for all a,bϵR we have
φ(a+b)=φ(a)+φ(b)
φ(ab)=φ(a)Xφ(b)
φ(1)=1 - 5KERNEL OF φSuppose R, R' are rings and φ:R->R' is a ring homomorphism. We define kerφ by kerφ={aϵR: φ(a)=0}
- 6IDEALLet R be a ring, I⊂R. We say I is an ideal of R if I is an additive subgroup of R so that for any aϵR, xϵI we have axϵI.
- 7QUOTIENT RINGLet I be an ideal of a ring R. Let R/I denote the set {x+I:xϵR} where x+I={x+a: aϵI}. R/I is called a quotient ring.
- 8COSETLet I be an ideal of a ring R. The elements of a quotient ring are called cosets of I.
- 9ISOMORPHISM/ ISOMORPHICA bijective homomorphism is called an isomorphism.
(If there is an isomorphism from a ring R onto a ring R', then we say R and R' and isomorphic, and we denote this by R≅R'). - 10ZERO DIVISORLet R be a ring. A non-zero element xϵR is called a zero divisor if there is some non-zero yϵR s.t. xy=0.
- 11INTEGRAL DOMAINA ring R≠{0} is called an integral domain if it has no zero divisors.
- 12PRIME IDEALA prime ideal of a ring R is an ideal I so that I≠R, and whenever abϵI (where a,bϵR), either aϵI or bϵI.
- 13FIELDA ring F is a field if 0≠1, and every non zero element of F has a multiplicative inverse.
- 14MAXIMAL IDEALAn ideal I of a ring is maximal if I≠R, and for any ideal J so that I⊆J⊆R, we have either J=I or J=R.
- 15FIELD OF FRACTIONSWith R an integral domain, define a relation ~ on (RX(R∖{0})) by (a,b)~(c,d) if ad=bc. Then ~ is an equivelance relation.
Letting a/b denote the equivalence class of (a,b) and Q={a/b:a,bϵR, b≠0}. Q is the field of fractions where
a/b+c/d=(ad+bc)/bd,
(a/b)X(c/d)=(ac)/(bd). - 16EUCLIDEAN DOMAINLet R be an integral domain. R is called a Euclidean Domain if there is a map δ:R\{0}->ℤ (≥0) so that:
(1) For any f,gϵR with f≠0, ∃q,rϵR s.t. g=fq+r with r=0 or δ(r)<δ(f)
(2) For any f,gϵR with f,g≠0, δ(fg)≥δ(f) - 17PRINCIPAL IDEALLet R be a ring and aϵR, the ideal generated by a is denoted by (a) and is equal to (a)={ax:xϵR}. We call such an ideal (a) a principal ideal.
- 18PRINCIPAL IDEAL DOMAIN (PID)An integral domain in which every ideal is principal is called a principal ideal domain.
- 19IRREDUCIBLELet R be an integral domain and take aϵR. We say a is irreducible if a is not 0 or a unit, and whenever a=cd then either c or d is a unit.
- 20ASSOCIATESFor irreducible a,bϵR, we say a and b are associates if a=bu, where u is a unit.
- 21UNIQUE FACTORISATION DOMAIN (UFD)Let R be an integral domain. Then we say R is a UFD if
(1) Every non-zero non-unit element of R canb e factored into a product of a finite number of irreducible elements; and
(2) The above factorisation is essentially unique, meaning: whenever p(1)...p(m)=q(1)...q(n) with p(i),q(j) irreducible, then m=n and after re-ordering the q(j), we have that for each j, p(j) and q(j) are associates. - 22HIGHEST COMMON FACTOR (HCF)Let R be a UFD, a,bϵR, not both 0. Set c=hcf(a,b) where cϵR so that c|a and c|b, and whenever dϵR so that d|a and d|b, we have d|c.
- 23RELATIVELY PRIMEWhen hcf(a,b)=1, we say a and b are relatively prime.
- 24CONTENTSuppose R is a UFD, and f=a(0)+a(1)X+...+a(n)X^(n)
ϵR[X] with f≠0.
We define the content of f to be hcf(a(0), a(1), ..., a(n)) - 25PRIMITIVEWe say fϵR[X} is primitive if f≠0 and the content of f is 1.
- 26MONICLet R be a ring, fϵR[x], f≠0. Let n=degf and write f=a(0)+a(1)X+...+a(n)X^(n)
[so a(n) ≠0],
We say f is monic if a(n)=1. - 27LEADING COEFFICIENTLet R be a ring, fϵR[x], f≠0. Let n=degf and write f=a(0)+a(1)X+...+a(n)X^(n)
[so a(n) ≠0],
a(n) is called the leading coefficient of f. - 28ROOTLet R, R' be rings with R⊆R'; fix αϵR'. Define homomorphism
φ(α):R[X]->R'[X] by φ(α)[c(0)+c(1)X+...c(n)X^(n)]=
c(0)+c(1)α+...c(n)α^(n). Let f(α) denote φ(α)(f). We say αϵR is a root of fϵR[X] if f≠0 and f(α)=0. - 29FIELD EXTENSIONSuppose K, L are fields and φ:K->L is a homomorphism (thus φ is necessarily injective). Then we say L is a field extension of K (or L:K is a field extension).
(When K,L are fields, K⊆L assume φ:K->L is identity map on K). - 30DEGREESuppose L:K is a field extension. Define the degree of L:K to be the dimension of L as a vector space over K. We use [L:K] to denote this. L:K is a finite extension if [L:K]<∞.
- 31ALGEBRAIC/ TRANSCENDENTALSay L:K is a field extension and αϵL. Identify K with its isomorphic image in L. We say α is algebraic over K if α is the root of some polynomial in K[X]. When α is not algebraic over K, we say α is transcendental over K. When every element of L is algebraic over K, we simply say L is algebraic over K.
- 32MINIMAL POLYNOMIALFor L:K a field extension with αϵL so that α is algebraic over K, the unique monic polynomial m(α)(K)ϵK[X] that generates a non-zero ideal of K[X], I={fϵK[X]:f(α)=0} is called the minimal polynomial of α over K.
- 33K[α]Let L:K be a field extension, αϵL. Let K[α] denote the smallest subring of L containing K and α.
- 34K(α)Let L:K be a field extension, αϵL. Let K(α) be the smallest subfield of L containing K and α.
- 35CHARACTERISTICLet K be a field with additive identity 0(K) and multiplicative identity 1(K). We write 2X1(K) to denote 1(K)+1(K) and 3X1(K) to denote 1(K)+1(K)+1(K) etc.
We define the characteristic of K, denoted charK, to be the smallest positive integer n so that nX1(K)=0(K); if no such n exists, we define the characteristic of K to be 0.
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- FlashCards sobre ALGEBRA DEFINITIONS, criado por artibv em 29-04-2014.
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