4) Factor Rings Flashcards
What is a Factor Ring / Quotient Ring (R/I)
In what sense are the operations (+) and (×) on the quotient ring R/I well-defined
Describe the proof that operations (+) and (×) on the quotient ring R/I are well-defined
What are the structural elements and inverse properties of the quotient ring R/I
- Ring Structure: The set R/I with operations
(+) and (×) defined by coset addition and multiplication forms a ring - Zero Element: the zero element of this ring is the coset 0 + I = I
- Identity Element: the identity element is 1 + I
- Multiplicative Inverse: −(r + I) = (−r) + I and, if r has an inverse r^1 in R then (r + I) is invertible in R/I with inverse r ^−1 + I
Describe the proof of the structural elements and inverse properties of the quotient ring R/I
What is the characteristic of the factor ring R/I if R has characteristic n ≠ 0
What are the key properties and implications of the canonical projection map π : R → R/I and the induced map from a homomorphism θ : R → S
What is the Fundamental Isomorphism Theorem
Explain the one-to-one correspondence between the ideals of a ring R containing a proper ideal I and the ideals of the factor ring R/I
Describe the isomorphism between quotient rings when one ideal is contained within another
What is a maximal ideal
An ideal I of a ring R is maximal if it is proper and for any ideal J with I ≤ J ≤ R, then J = I or J = R
What is the relationship between a maximal ideal in a commutative ring and the properties of the corresponding quotient ring
What is a prime ideal
A proper ideal I of a commutative ring R is prime if whenever r, s ∈ R and rs ∈ I then either r ∈ I or s ∈ I
When is a proper ideal in a commutative ring a prime ideal
Let I be a proper ideal of a commutative ring R. Then I is a prime ideal iff R/I is an integral domain
What is a the relationship between prime, integral domain, maximal and field
R/I field <=> I maximal
R/I field => R/I Integral Domain
R/I integral domain <=> I Prime
