2.3 Construction and properties of quotient rings

Quotient rings are a powerful tool in algebra, letting us create new rings from existing ones. They're formed by taking a ring and an ideal, then grouping elements that differ by the ideal into equivalence classes.

The construction of quotient rings involves defining addition and multiplication on these classes. This process preserves key ring properties while potentially introducing new algebraic behaviors, like zero divisors or field structures.

Definition and Construction of Quotient Rings

Definition of quotient rings

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• Quotient ring R/I forms set of cosets of I in R where I is an ideal of R
• Coset of element r in R defined as $r + I = \{r + i : i \in I\}$ representing equivalence class
• Elements a, b in R considered equivalent if $a - b \in I$ establishing equivalence relation
• R/I notation read as "R mod I" or "R modulo I" signifying modular arithmetic
• I being an ideal ensures well-defined operations in R/I preserving algebraic structure

Construction of R/I

• Elements of R/I comprise cosets of I in R represented as $[r]$ or $r + I$ for $r \in R$
• Addition in R/I defined as $[a] + [b] = [a + b]$ preserving algebraic structure
• Multiplication in R/I defined as $[a] \cdot [b] = [ab]$ maintaining ring properties
• Zero element identified as $[0] = I$ serving as additive identity
• Additive inverse of $[a]$ given by $-[a] = [-a]$ ensuring group structure
• Construction process yields new algebraic structure from existing ring and ideal

Proof of ring structure

• Well-defined operations demonstrated through:
  1. Addition: $[a_1] = [a_2]$ and $[b_1] = [b_2]$ imply $[a_1 + b_1] = [a_2 + b_2]$
  2. Multiplication: $[a_1] = [a_2]$ and $[b_1] = [b_2]$ imply $[a_1b_1] = [a_2b_2]$
• Ring axioms verified:
  1. Closure under addition and multiplication
  2. Associativity of addition and multiplication
  3. Commutativity of addition
  4. Existence of additive identity $[0]$
  5. Existence of additive inverses
  6. Distributivity of multiplication over addition
• Proof establishes R/I as legitimate ring structure

Properties of quotient rings

• Commutativity of R/I inherited from R preserving multiplication order
• Unity $[1]$ in R/I exists if R has unity 1 acting as multiplicative identity
• Zero divisors may emerge in R/I even if R lacks them related to elements not in I
• R/I forms integral domain if and only if I is prime ideal of R ensuring no zero divisors
• R/I becomes field if and only if I is maximal ideal of R yielding multiplicative inverses
• Characteristic of R/I determined by characteristic of R and elements in I affecting algebraic behavior