Algebraic Structures Exam 2

Front 1

Fundamental Theorem of Finite Abelian Groups

Back 1

Every finite Abelian group is a direct product of cyclic groups of prime-power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group

Front 2

Ring

Back 2

A ring R is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), such that for all a, b,
c in R:
1. a + b = b + a.
2. (a + b) + c = a + (b + c).
3. There is an additive identity 0. That is, there is an element 0 in R such that a + 0 = a for all a in R.
4. There is an element -a in R such that a + (-a) = 0.
5. a(bc) = (ab) c.
6. a(b + c) = ab + ac and (b + c)a = ba + ca.

Front 3

Unity (or identity)

Back 3

In a ring, a unity is a nonzero element that is an identity under multiplication.

Front 4

Unit

Back 4

A nonzero element of a commutative ring with unity that has a multiplicative inverse. That is, if a is a unit then a-1 exists.

Front 5

Uniqueness of the Unity and Inverses

Back 5

If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.

Front 6

Subring

Back 6

A subset S of a ring R is a subring of R if S is itself a ring with the operations of R.

Front 7

Subring Test

Back 7

A nonempty subset S of a ring R is a subring if S is closed under subtration and multiplication -- that is, if a - b and ab are in S
whenever a and b are in S.

Front 8

Zero-Divisors

Back 8

A zero-divisor is a nonzero element a of a commutative ring R such that there is a nonzero element b e R with ab = 0.

Front 9

Integral Domain

Back 9

An integral domain is a commutative ring with unity and no zero-divisors.

Front 10

Cancellation

Back 10

Let a, b, and c belong to an integral domain. If a ∫ 0 and ab = ac, then b = c.

Front 11

Field

Back 11

A field is a commutative ring with unity in which every nonzero element is a unit.

Front 12

Finite Integral Domains Are Fields

Back 12

A finite integral domain is a field.

Front 13

Characteristic of a Ring

Back 13

The characteristic of a ring R is the least positive integer n such that 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 by char R.

Front 14

Ideal

Back 14

A subring A of a ring R is called a (two-sided) ideal of R if for every r e R and every a e A both ra and ar are in A

Front 15

Ideal Test

Back 15

A nonempty subset A of a ring R is an ideal of R if
1. a - b e A whenever a, b e A.
2. ra and ar are in A whenever a e A and r e R.

Front 16

Existance of Factor Rings

Back 16

Let R be a ring and let A be a subring of R. The set of cosets {r + A | r e R} is a ring under the operations (s + A) + (t + A) = s + t + A and (s + A)(t + A) = st + A if and only if A is an ideal of R.

Front 17

Prime Ideal, Maximal Ideal

Back 17

A prime ideal A of a commutative ring R is a proper ideal of R such that a, b e R and ab e A imply a e A or b e A. A maximal
ideal of a commutative ring R is a proper ideal of R such that, whenever B is an ideal of R and A ΠB ΠR, then B = A or B = R.

Front 18

R/A is an Integral Domain If and Only If A is Prime

Back 18

Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is an integral domain if and only if A is prime.

Front 19

Ring Homomorphism, Ring Isomorphism

Back 19

A ring homomorphism f from a ring R to a ring S is a mapping from R to S that preserves the two ring operations; that is, for all
a, b in R,
f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).
A ring homomorphism that is both one-to-one and onto is called a ring isomorphism

Front 20

First Isomorphism Theorem for Rings!

Back 20

Let f be a ring homomorphism from a ring R to a ring S. Then the mapping from R/Ker f to f(R), given by r + Ker f Ø f(r), is
an isomorphism. In symbols, R/Ker f º f(R).

Front 21

Ring of Polynomials over R

Back 21

Let R be a commutative ring. The set of formal symbols
R[x] = 8an xn+ an-1 xn-1+ --- + a1 x + a0 | ai œ R, n is a nonnegative integer} is called the ring of polynomials over R in the indeterminate x (remember, x is NOT a variable). Two elements an x n + an-1 x n-1+ --- + a1 x + a0 and bm xm+ bm-1 x m-1+ --- + b1 x + b0 of R[x] are considered equal if and only if ai = bi for all nonnegative integers i. (Define ai = 0 when i > n and bi = 0 when i > m)

Front 22

D an Integral Domain Implies, D[x] an Integral Domain

Back 22

If D is an integral domain, then D[x] is an integral domain

Front 23

Division Algorithm for F[x]

Back 23

Let F be a field and let f(x) and g(x) œ F[x] with g(x) ∫ 0. Then there exists 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).

Front 24

Irreducible Polynomial, Reducible Polynomial

Back 24

Let D be an integral domain. A polynomial f(x) from D[x] that is neither the zero polynomial nor the unit in D[x] is said to be
irreducible over D if, whenever f(x) is expressed as a product f(x) = g(x)h(x), with g(x) and h(x) from D[x], then g(x) or h(x) is a
unit in D[x]. A nonzero, nonunity element of D[x] that is not irreducible over D is called reducible over D.

Front 25

Reducibility over Q Implies Reducibility Over Z

Back 25

Let f(x) œ F[x]. If f(x) is reducible over Q, then it is reducible over Z.

Front 26

Mod p Irreducibility Test

Back 26

Let p be a prime and suppose that f(x) œ Z[x] with deg f(x) ¥ 1. Let f
-
(x) be the polynomial in Zp[x] obtained from f(x) by
reducing all the coefficients of f(x) modulo p. If f
-
(x) is irreducible over Zp and deg f
-
HxL =deg f(x), then f(x) is irreducible over
Q.

Front 27

Eisentsein's Criterion (1850)

Back 27

Let f(x) = an x n+ an-1 xn-1+ --- + a1 x + a0 œ Z[x]
If there is a prime p such that p I an, p ˝ an-1, ... , p ˝ a0 and p
2 I a0, then f(x) is irreducible over Q.

Front 28

F[x]/Xp(x)\ Is a Field

Back 28

Let F be a field and p(x) an irreducible polynomial over F. Then F[x]/Xp(x)\ is a field.

Front 29

Associates, Irreducibles, Primes

Back 29

Elements a and b of an integral domain D are called associates if a = ub, where u is a unit of D. A nonzero element a of an
integral domain D is called an irreduxible if a is not a unit and, whenever b, c œ D with a = bc, then b or c is a unit. A nonzero
element a of an integral domain D is called prime if a is not a unit and a | bc implies a | b or a | c.

Front 30

Euclidean Domain

Back 30

An integral domain D is called a Euclidean domain if there is a function d (called the measure) from the nonzero elements of D
to the nonnegative integers such that
1. d(a) § d(ab) for all nonzero a, b in D; and
2. if a, b œ D, b ∫ 0, then there exist elements q and r in D such that a = bq + r, where r = 0 or d(r) < d(b).

Front 31

Corollary: Existance of Subgroups of Abelian Groups

Back 31

If m divides the order of a finite Abelian group G, then G has a subgroup of order m.

Front 32

Characteristic of a Ring with Unity

Back 32

Let R be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of R is 0. If 1 has order n under addition, then the characteristic of R is n.

Front 33

Characteristic of an Integral Domain

Back 33

The characteristic of an integral domain is 0 or prime.

Front 34

R/A is a Field If and Only If A is Maximal

Back 34

Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is a field if and only if A is maximal.

Front 35

Kernels Are Ideals
Ideals Are Kernels

Back 35

Let f be a ring homomorphism from a ring R to a ring S. Then Ker f = {r œ R | f(r) = 0} is an ideal of R.
Every ideal of a ring R is the kernel of a ring homomorphism of R. In particular, an ideal A is the kernel of the mapping r Ø r + A from R to R/A.

Front 36

Corollary 1: A Ring with Unity Contains Zn or Z

Back 36

If R is a ring with unity and the characteristic of R is n > 0, then R contains a subring isomorphic to Zn. If the characteristic of R
is 0, then R contains a subring isomorphic to Z.

Front 37

Corollary 3: A Field Contains Zp or Q (Steitz, 1910)

Back 37

If F is a field of characteristic p, then F contains a subfield isomorphic to Zp. If F is a field of characteristic 0, then F contains a
subfield isomorphic to the rational numbers.

Front 38

Field of Quotients

Back 38

Let D be an integral domain. Then there exists a field F (called the field of quotients of D) that contains a subring isomorphic to
D.

Front 39

D an Integral Domain Implies, D[x] an Integral Domain

Back 39

If D is an integral domain, then D[x] is an integral domain.

Front 40

F[x] is a PID

Back 40

Let F be a field. Then F[x] is a principal ideal domain.

Front 41

Criterion for I = <g(x)>

Back 41

Let F be a field, I a nonzero ideal in F[x], and g(x) an element of F[x]. Then, I = < g(x) > if and only if g(x) is a nonzero polynomial of minimum degree in I.

Front 42

Prime Implies Irreducible

Back 42

In an integral domain, every prime is an irreducible.