Abstract Algebra Exam 2 Studying

Kernel of a Group Homomorphism (Definition 2.41)

Let φ: G → G' be a group homomorphism. The kernel of φ is the set of elements of G that are sent to the identity e' of G' by φ: ker(φ) = {g ∈ G | φ(g) = e'}.

Ring (Definition 3.1)

A ring R is a set with two operations, addition (written "+") and multiplication (written "·") satisfying the following axioms: (1) The set R with its addition law + is an abelian group. Denote the identity of this group by 0 or 0R. (2) The multiplication law · is associative, i.e., a · (b · c) = (a · b) · c for all a, b, c ∈ R, and there is an element 1R ∈ R satisfying 1R · a = a · 1R = a for all a ∈ R. (3) For all a, b, c ∈ R we have a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a. If it is further true that a · b = b · a for all a, b ∈ R, then the ring is said to be commutative.

Ring Homomorphism (Definition 3.3)

Let R and R' be rings. A ring homomorphism from R to R' is a function φ: R → R' satisfying φ(1R) = 1R', φ(a + b) = φ(a) + φ(b) for all a, b ∈ R, φ(a · b) = φ(a) · φ(b) for all a, b ∈ R.

Kernel of a Ring Homomorphism (Definition 3.3)

If φ: R → R' is a ring homomorphism, the kernel of φ is the set of elements that are sent to 0R' by φ: ker(φ) = {a ∈ R | φ(a) = 0R'}.

Ring Isomorphism (Definition 3.3)

Let R and R' be rings. A function φ: R → R' is an isomorphism if it is a bijective ring homomorphism. In this case, we say R and R' are isomorphic, and we write R ≅ R'.

Field (Definition 3.10)

A field is a commutative ring R with the property that every non-zero element of R has a multiplicative inverse. In other words, for every a ∈ R with a ≠ 0 there is a b ∈ R satisfying ab = 1.

Zero Divisor (Definition 3.13)

Let R be a commutative ring. An element a ∈ R is called a zero divisor if a ≠ 0 and there is some non-zero b ∈ R such that ab = 0.

Integral Domain (Definition 3.13)

A ring R is an integral domain if it has no zero divisors.

Cancellation Property (Proposition 3.15)

A commutative ring R has the cancellation property if for every a, b, c ∈ R, we have ab = ac if and only if b = c or a = 0.

Group of Units (Definition 3.16)

Let R be a commutative ring. The group of units of R is the subset R* of R defined by R* = {a ∈ R | there is some b ∈ R satisfying ab = 1}. The group law on R* is ring multiplication. The elements of R* are called the units of R.

Product of Rings (Definition 3.22)

Let R1, . . . , Rn be rings. The product of R1, . . . , Rn is the ring R1 × · · · × Rn = {(a1, . . . , an) | ai ∈ Ri}, with multiplication and addition defined component-wise.

Ideal of a Ring (Definition 3.26)

Let R be a commutative ring. An ideal of R is a nonempty subset I ⊆ R satisfying:

• If a ∈ I and b ∈ I, then a + b ∈ I (closure under addition).

• If a ∈ I and r ∈ R, then ra ∈ I (closure under multiplication by elements in R).

Principal Ideal (Definition 3.27)

Let R be a commutative ring, and let c ∈ R. The principal ideal generated by c, denoted cR or (c), is the set of all multiples of c, cR = (c) = {rc | r ∈ R}.

I-coset of a (Definition 3.31)

Let R be a commutative ring, and let I be an ideal of R. For each a ∈ R, the I-coset of a is the set a + I = {a + c | c ∈ I}.

Coset Operations (Definition 3.31)

Let R be a commutative ring, and let I be an ideal of R. Given two cosets a+I and b + I, we define their sum and product by the formulas (a + I) + (b + I) = (a + b) + I, (a + I) · (b + I) = (a · b) + I.

Quotient Ring (Definition 3.31)

Let R be a commutative ring, and let I be an ideal of R. The quotient ring R/I is the set of I-cosets endowed with addition and multiplication of cosets using the formulas (a + I) + (b + I) = (a + b) + I, (a + I) · (b + I) = (a · b) + I.

Prime Ideal (Definition 3.37)

Let R be a commutative ring. An ideal I of R is a prime ideal if I ≠ R and if whenever a product of elements ab ∈ I, then either a ∈ I or b ∈ I.

Maximal Ideal (Definition 3.40)

Let R be a commutative ring. An ideal I is called a maximal ideal if I ≠ R and if there are no ideals properly contained between I and R. In other words, if J is an ideal of R and I ⊆ J ⊆ R, then J = I or J = R.

F-vector Space (Definition 4.3)

Let F be a field. An F-vector space is an abelian group (V, +) with a rule for multiplying a vector v in V by a scalar c in F to obtain a new vector cv in V.

F-Vector Space Axioms

(1) (Identity Law) 1v = v for all v ∈ V.

(2) (Distributive Law #1) c(v1 + v2) = cv1 + cv2 for all v1, v2 ∈ V and all c ∈ F.

(3) (Distributive Law #2) (c1 + c2)v = c1v + c2v for all v ∈ V and all c1, c2 ∈ F.

(4) (Associative Law) (c1c2)v = c1(c2v) for all v ∈ V and all c1, c2 ∈ F.

The identity element of V is called the zero vector and is denoted by 0.

Subspace of V (Definition 10.19)

Let V be an F-vector space. A subspace of V is a nonempty subset U ⊆ V satisfying the following two properties:

(1) (Closure under addition) If v, w ∈ U, then v + w ∈ U.

2) (Closure under scalar multiplication) If v ∈ U and c ∈ F, then cv ∈ U.

Linear Transformation (Definition 4.15)

Let F be a field, and let V and W be F-vector spaces. A linear transformation from V to W is a function L: V → W satisfying L(c1v1 + c2v2) = c1L(v1) + c2L(v2) for all v1, v2 ∈ V and all c1, c2 ∈ F.

Finite Basis for V (Definition 4.11)

Let V be an F-vector space. A finite basis for V is a finite set of vectors B = {v1, . . . , vn} ⊆ V with the following property: Every vector v ∈ V can be written in the form v = a1v1 + a2v2 + · · · + anvn for exactly one choice of scalars a1, . . . , an ∈ F.

Spanning Set (Definition 4.14)

Let V be an F-vector space, and let A = {v1, . . . , vn} be a finite set of vectors in V. The set A spans V if every vector in V is a linear combination of the vectors in A.

Linear Independence (Definition 4.14)

Let V be an F-vector space, and let A = {v1, . . . , vn} be a finite set of vectors in A. The set A is linearly independent if the only scalars that make a1v1 + a2v2 + · · · + anvn = 0 are a1 = a2 = · · · = 0.

Span of A (Definition 4.14)

Let V be an F-vector space, and let A = {v1, . . . , vn} be a set of vectors in V. Then the span of A, denoted Span(A), is the set of linear combinations of elements of A. That is, Span(A) = {v1, . . . , vn}.

Basis and Spanning/Linear Independence (Proposition 4.15)

Let V be an F-vector space, and let A = {v1, . . . , vn} be a set of vectors in V. Then A is a basis for V if and only if A both spans V and is linearly independent.

Goldilocks Theorem (Theorem 4.16)

Let V be an F-vector space, let S be a finite set of vectors in V that spans V, and let L ⊆ S be a subset of S that is linearly independent. Then there is a basis B for V satisfying L ⊆ B ⊆ S.

Invariance of Dimension (Theorem 4.18)

Let V be a vector space that has a finite basis. Then every basis for V has exactly the same number of elements.

Dimension of V (Definition 4.19)

Let V be a vector space. If V has a finite basis, the dimension of V is the number of vectors in a basis of V.