Glossary

2.3 Construction and properties of quotient rings

2 min read•july 25, 2024

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 , 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 structures.

Definition and Construction of Quotient Rings

Definition of quotient rings

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  • forms set of cosets of I in R where I is an ideal of R
  • of element r in R defined as r+I={r+i:i∈I}r + I = \{r + i : i \in I\} representing
  • Elements a, b in R considered equivalent if a−b∈Ia - b \in I establishing
  • 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][r] or r+Ir + I for r∈Rr \in R
  • Addition in R/I defined as [a]+[b]=[a+b][a] + [b] = [a + b] preserving algebraic structure
  • Multiplication in R/I defined as [a]⋅[b]=[ab][a] \cdot [b] = [ab] maintaining ring properties
  • identified as [0]=I[0] = I serving as
  • of [a][a] given by −[a]=[−a]-[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: [a1]=[a2][a_1] = [a_2] and [b1]=[b2][b_1] = [b_2] imply [a1+b1]=[a2+b2][a_1 + b_1] = [a_2 + b_2]
    2. Multiplication: [a1]=[a2][a_1] = [a_2] and [b1]=[b2][b_1] = [b_2] imply [a1b1]=[a2b2][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][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
  • [1][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 if and only if I is of R ensuring no zero divisors
  • R/I becomes field if and only if I is of R yielding multiplicative inverses
  • Characteristic of R/I determined by characteristic of R and elements in I affecting algebraic behavior

Key Terms to Review (19)

Addition in quotient rings: Addition in quotient rings is the operation of combining equivalence classes of elements from a ring, where the elements belong to the same equivalence class if they differ by an element of a given ideal. This process highlights how we can treat sets of elements as single entities, facilitating the study of ring properties through these simplified structures.
Additive Identity: The additive identity is the unique element in a mathematical structure, typically a set with an addition operation, that, when added to any other element in that structure, leaves the other element unchanged. In most number systems, this element is zero. Understanding the additive identity is crucial in the context of operations involving quotient rings, as it helps establish the foundational properties of these algebraic structures.
Additive Inverse: The additive inverse of an element in a mathematical structure is the element that, when added to the original element, yields the additive identity, which is typically zero. Understanding the concept of additive inverses is crucial when dealing with algebraic structures like rings and fields, particularly in the construction and properties of quotient rings, as they help to establish essential operations such as subtraction within these structures.
Coset: A coset is a set formed by multiplying all elements of a subgroup by a fixed element from the larger group. It plays a critical role in understanding the structure of groups and is fundamental when working with quotient structures, especially in the context of constructing and analyzing quotient rings.
Equivalence Class: An equivalence class is a subset of a set formed by grouping together elements that are considered equivalent under a specific relation. This concept is crucial in understanding how sets can be partitioned into distinct groups where all elements within each group share a common property defined by the equivalence relation. In the context of rings, this concept helps in forming quotient rings where elements that differ by a specific ideal are treated as equivalent.
Equivalence Relation: An equivalence relation is a binary relation that partitions a set into disjoint subsets, called equivalence classes, where each element is related to itself, each element is related to another element if they are considered equivalent, and the relation is symmetric. This concept is fundamental in abstract algebra as it helps to define structures like quotient rings and quotient modules, as well as the process of localization, providing a way to group elements that share certain properties.
Field: A field is a set equipped with two operations, addition and multiplication, that satisfy certain properties, allowing for the division of non-zero elements. Fields play a crucial role in algebra since they provide a structure where every non-zero element has a multiplicative inverse, making them essential in understanding commutative rings and integral domains. The properties of fields enable operations such as finding quotients and establishing isomorphisms between algebraic structures.
First Isomorphism Theorem: The first isomorphism theorem states that if there is a ring homomorphism from a ring $R$ to a ring $S$, then the image of this homomorphism is isomorphic to the quotient of $R$ by the kernel of the homomorphism. This theorem connects the concepts of homomorphisms, kernels, and quotient rings, illustrating how structure can be preserved through mappings between rings.
Ideal: An ideal is a special subset of a ring that is closed under addition and absorbs multiplication by any element from the ring. Ideals play a critical role in understanding the structure of rings, allowing us to construct quotient rings and study their properties. They also help in defining concepts like homomorphisms and isomorphisms, which are essential for analyzing relationships between different rings.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors, meaning that the product of any two non-zero elements is always non-zero. This property ensures that integral domains have certain arithmetic characteristics similar to those of integers, making them foundational in the study of algebraic structures.
Maximal ideal: A maximal ideal is an ideal in a ring that is proper (not equal to the entire ring) and has the property that there are no other ideals containing it except for itself and the entire ring. These ideals play a crucial role in understanding the structure of rings, especially in relation to fields and quotient rings.
Multiplication in Quotient Rings: Multiplication in quotient rings involves the process of multiplying equivalence classes of elements from a ring with respect to a particular ideal. This operation is well-defined and respects the ring structure, meaning that the product of two equivalence classes yields another equivalence class, following specific rules dictated by the ideal. The properties of these products play a crucial role in understanding how quotient rings function and interact with the original ring.
Prime Ideal: A prime ideal in a commutative ring is a proper ideal such that if the product of two elements is in the ideal, at least one of those elements must also be in the ideal. This concept is essential for understanding the structure of rings and has deep connections with other algebraic concepts, including maximal ideals and quotient rings. Prime ideals play a crucial role in defining prime elements and their relationship to irreducibility in algebraic structures.
Quotient Ring: A quotient ring is a type of ring formed by taking a commutative ring and dividing it by one of its ideals. It provides a way to construct new rings from existing ones, revealing important algebraic structures and properties. Quotient rings are crucial in understanding the relationships between rings and their ideals, especially in the context of isomorphism theorems and the characterization of prime and maximal ideals.
R/i: In the context of algebra, r/i represents a quotient ring formed by taking a ring r and an ideal i. It captures the concept of creating a new ring where the elements of the ideal i are treated as equivalent to zero, essentially 'collapsing' those elements in r. This construction allows for simplified analysis and manipulation of algebraic structures while preserving important properties of the original ring.
Second Isomorphism Theorem: The second isomorphism theorem states that if you have a ring and a subring that is also an ideal, then the quotient of the ring by this ideal is isomorphic to the quotient of the subring by its intersection with the ideal. This theorem highlights the relationship between subrings and their ideals, as well as how they interact within larger structures like rings and modules.
Unity: Unity, in the context of algebra, refers to the multiplicative identity element, commonly denoted as 1. It is the element in a ring such that when multiplied by any other element in that ring, it leaves the other element unchanged. Understanding unity is crucial in exploring the properties and structure of quotient rings, where it plays a key role in defining equivalence classes and ensuring that the ring remains closed under multiplication.
Zero Divisor: A zero divisor is an element in a ring that, when multiplied by another non-zero element, yields zero. This concept is crucial in understanding the structure of rings and their properties, particularly when exploring quotient rings where zero divisors can influence the behavior of elements and the formation of ideals. Recognizing the role of zero divisors helps to clarify the conditions under which certain mathematical operations are valid and how they affect the algebraic framework.
Zero Element: The zero element, often denoted as 0, is a fundamental component in algebraic structures like rings and groups, serving as the additive identity. This means that when you add the zero element to any other element in the structure, the value remains unchanged, which is essential for defining operations within quotient rings. Understanding the role of the zero element helps in grasping how quotient rings are constructed and their properties, particularly in relation to equivalence classes.
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