CCE2 Flashcards
Homomorphism
For rings R and R
a map φ : R → R is a homomorphism if the following two conditions are satisfied for all a, b ∈ R:
1. φ(a + b) = φ(a) + φ(b),
2. φ(ab) = φ(a)φ(b).
An isomorphism
R → R’ from a ring R to a ring R’ is a homomorphism that is one-to-one and onto R’. The rings R and R’ are then isomorphic.
Describe all ring homomorphisms of Z into Z.
Let φ : Z→Z be a ring homomorphism. Because 1^2= 1, we see that φ (1) must be
an integer whose square is itself, namely either 0 or 1. If φ(1) 1 = then
φ (n)=φ ( n.1) so φ is the identity map of Z onto itself, which is a homomorphism. If φ (1) =0 then φ (n.1)=0 so φ maps everything onto 0, which also yields a homomorphism.
- Show that if D is an integral domain, then {n · 1 | n ∈ Z} is a subdomain of D contained in every subdomain
of D.
Let R={n.1|n∈Z} We have n.1+m.1=(n+m).1 so R is closed under addition. Taking n=0 we see that 0∈ R. Because the inverse of n.1 is (-n).1, We see that R contains all additive inverses of elements, SO <R, +> is an abelian group. The distributive laws show that (n . 1)(m 1) = (nm).1, so R is closed under multiplication. Because 1.1 =1, we see that 1∈ R. Thus, R is a commutative ring with unity. Because a product ab=0 in R can also be viewed as a product in D, we see that R also has no divisors of zero. Thus, R is a subdomain of D.
What is the field of quotients of an integral domain?
The field of quotients (or field of fractions) of an integral domain D is the set of all elements of the form a/b and a,b∈D, and 𝑏≠0. This set forms a field 𝐹 where:
add: a/b+c/d=(ad+bc)/bd
mul: a/b*c/d=ac/bd D is embedded in F by identifying each a∈D with a/1. this allows division by any non-zero element of D, forming a field
An integral domain
D is a commutative ring with unity 1 ̸= 0 that contains no divisors of 0.
Every finite integral domain is a field.
Every finite integral domain is a field.
What is the characteristic of a ring?
For a ring R, the characteristic is the smallest positive integer n such that n.a=0 for all a∈R where n.a means a adding itself n times. if no n exists, the ring has characteristic 0.
What is the characteristic of a field?
A field has a characteristic of either 0 or a prime p. If the characteristic is 0, no integer multiple of 1 equals 0. If the characteristic is p, then
𝑝⋅1=0 for all elements in the field.
What is the characteristic of an integral domain?
The characteristic of an integral domain is either 0 or a prime number p. If the characteristic were composite, it would introduce zero divisors, which contradicts the definition of an integral domain.
Show that the characteristic of an integral domain D must be either 0 or a prime p. [Hint: If
the characteristic of D is mn, consider (m · 1)(n · 1) in D.]
Suppose the characteristic is mn for m > 1 and n > 1. Following the hint, the distributive laws show that (m · 1)(n · 1) = (nm) ·1 = 0. Because we are in an integral domain, we must have either m · 1 = 0 or n ·1=0. But if m · 1 = 0 then Theorem 23.14 shows that the characteristic of D is at most m. If n · 1 = 0, the characteristic of D is at most n. Thus the characteristic can’t be a composite positive integer, so it must either be 0 or a prime p.
Properties of Homomorphisms #1
- If 0 is the additive identity in R, then φ(0) = 0 is the additive identity in R
- If a ∈ R, then φ(−a) = −φ(a).
- If S is a subring of R, then φ[S] is a subring of R’
- If S’ is a subring of R’, then φ−1[S] is a subring of R
Properties of homomorphism #2
- If R has unity 1, then φ(1) is unity for φ[R].
- If N is an ideal in R, then φ[N] is an ideal in φ[R].
- If N’ is an ideal in either R’ or φ[R], then φ−1[N’] is an ideal in R.
An isomorphism of a ring R with a ring R′ is a homomorphism φ : R →R′mapping R onto R′ such that Ker(φ ) {0}.
An isomorphism of a ring R with a ring R′ is a homomorphism φ : R →R′mapping R onto R′ such that Ker(φ ) {0}.
The kernel of a homomorphism φ mapping a ring R into a ring R′ is {r∈R|φ(r)=0’}.
The kernel of a homomorphism φ mapping a ring R into a ring R′ is {r∈R|φ(r)=0’}.
