Abstract Algebra: Ideals, Groups

Front 1

Define ideal and the structure in which it may occur

Back 1

A SUBRING I of a RING R is an ideal if for every r in R and every a in I,
ar and ra are BOTH in I.

Front 2

Name two conditions that are sufficient to show that a nonempty subset I is an ideal in a ring R.

Back 2

1- I is closed under subtraction.

2- I absorbs on the left and right under multiplication.

Front 3

Define a principal ideal and name the properties of the ring R in which it may occur.

Back 3

R must be COMMUTATIVE WITH IDENTITY. If c is in R, the principal ideal generated by c is the set of all multiples of c in R.

Front 4

Define a finitely generated ideal and name the properties of the ring R in which it may occur.

Back 4

R must be COMMUTATIVE WITH IDENTITY. If c_1,...,c_n are in R, the principal ideal generated by c_1,...,c_n is the set of all linear combinations of c_1,...,c_n in R.

Front 5

What does a≡b (mod I) mean, where I is an ideal?

Back 5

a-b is an element of I. This is a valid relation in ANY ring.

Front 6

What is the congruence class of a modulo I, if I is an ideal in a ring R and a is an element of R?

Back 6

a+I is the set of all elements b in R such that b≡a (mod I).

Front 7

If I is an ideal in a ring R, what is R/I and what properties does it have?

Back 7

R/I is the set of all congruence classes modulo I; that is, the set of all sets of the form a+I={a+i | i is an element of I}.
R/I is the QUOTIENT RING and is indeed a ring. If R is commutative, so is R/I. If R has identity, so does R/I.

Front 8

What is true of the kernel of a ring homomorphism that maps R to S?

Back 8

The kernel is an ideal in R. If the kernel is {0_R} only, then the function is injective.

Front 9

State the FIRST ISOMORPHISM THEOREM.

Back 9

Let f:R→S be a surjective homomorphism of rings with kernel K. Then R/K is isomorphic to S.

Front 10

Define a prime ideal and the kind of ring in which it may exist.

Back 10

An ideal P in a COMMUTATIVE ring R is prime if P≠R and whenever bc is in P, then b is in P or c is in P.

Front 11

Define a maximal ideal and the kind of ring in which it may exist.

Back 11

An ideal M in a COMMUTATIVE ring R is maximal if M≠R and whenever J is an ideal such that M is a subset of J and J is a subset of R, then either M=J or J=R.

Front 12

What is known about a Q, a quotient ring mod P, where P is a prime ideal? What if P is maximal?

Back 12

If P is prime, Q is an integral domain.
If P is maximal, Q is a field.

Front 13

Define permutation of a set T.

Back 13

A permutation of a set T is a bijective function f:T→T.

Front 14

Name the four group axioms.

Back 14

A NONEMPTY set G with operation * is a group if when a,b,c are in G:
1- Closure: a*b is in G
2- Associativity: (a*b)*c=a*(b*c)
3- Identity: there exists an element e in G such that a*e=e*a=a
4- Inverse: For every a, there exists a d in G such that a*d=d*a=e.

Front 15

What property is unique to abelian groups?

Back 15

Commutativity: a*b=b*a for all a,b in the group.

Front 16

What is another name for the set S_k of permutations on a set of k elements, where k is a finite positive integer? What is its order?

Back 16

The SYMMETRIC GROUP on k symbols. It is a nonabelian group of order k!.

Front 17

What is the name of the group of rotations and reflections of a k-sided polygon? What is its order?

Back 17

The DIHEDRAL group of degree k. It has order 2k.

Front 18

Why is a ring under multiplication not a group?

Back 18

Because zero has no multiplicative inverse.

Front 19

If R is a ring with identity, what is true of the set of all units in R?

Back 19

They form a group under multiplication.

Front 20

If G has order g and H has order h, what is the order of G X H?

Back 20

The order is gh.

Front 21

Under what conditions is cancellation valid in a group?

Back 21

Always.

Front 22

What is the order of an element a in a group?

Back 22

The SMALLEST POSSIBLE integer n such that a^n=e.

Front 23

If a has order n and a^k=e, what is the relation between k and n?

Back 23

n|k

Front 24

If a has order n and a^i=a^j, what is the relation between i and j?

Back 24

i≡j(mod n)

Front 25

If a has order n and n=td with d>0, what is the order of a^t?

Back 25

d

Front 26

What two conditions confirm that a subset of a group is a subgroup?

Back 26

Closure under the group operation and existence of the inverse element of every element in the set.

Front 27

What one condition confirms that a subset of a finite group is a subgroup?

Back 27

Closure under the group operation.

Front 28

What is the center of a group?

Back 28

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

Front 29

If G is a nonabelian group and a is in G, is the cyclic subgroup generated by a nonabelian?

Back 29

No.

Front 30

State CAYLEY'S THEOREM.

Back 30

Every group G is isomorphic to a group of permutations.

Front 31

If K is a subgroup of a group G and a,b are in G, what does it mean to say a≡b(mod K)?

Back 31

a(b^-1) is an element of K.

Front 32

If H is a subgroup of a group G, what is [G:H]?

Back 32

The index of H in G; that is, the number of distinct right cosets of H.

Front 33

State LAGRANGE'S THEOREM.

Back 33

If K is a subgroup of a FINITE group G, |G|=|K|[G:K], so |K| divides |G|.

Front 34

If a group G with element a has order g, what is known of the order of a?

Back 34

The order of a divides g and a^g=e.