2.1 Rings, ideals, and modules

Rings, ideals, and modules form the backbone of commutative algebra. These structures help us understand algebraic relationships and properties in mathematics, providing tools to analyze complex systems and solve equations.

By studying rings, ideals, and modules, we gain insight into fundamental algebraic concepts. These structures allow us to generalize familiar ideas from number theory and linear algebra, opening doors to advanced mathematical thinking and problem-solving techniques.

Rings, Ideals, and Modules

Definition and Examples of Rings

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• A ring is a set equipped with two binary operations, typically called addition and multiplication, satisfying axioms such as associativity, distributivity, and the existence of additive identity and inverses
• Examples of rings include the integers (ℤ), the rational numbers (ℚ), the real numbers (ℝ), the complex numbers (ℂ), and polynomial rings (ℝ[x])

Definition and Examples of Ideals

• An ideal is a subset of a ring closed under addition and absorption, meaning multiplying an element of the ideal by any element of the ring yields an element of the ideal
• Examples of ideals include the even integers in ℤ, the set of polynomials with constant term 0 in ℝ[x], and the principal ideal generated by a single element

Definition and Examples of Modules

• A module is a generalization of a vector space, where the scalars come from a ring instead of a field, consisting of an abelian group and a scalar multiplication operation satisfying certain axioms
• Examples of modules include vector spaces over fields (ℝ^n over ℝ), abelian groups (as modules over ℤ), and ideals in a ring (as modules over the ring)

Properties of Rings, Ideals, and Modules

Properties of Ideals

• Prove the intersection of any collection of ideals in a ring is also an ideal
• Show the sum of two ideals in a ring is an ideal
• Demonstrate the product of two ideals in a commutative ring is an ideal
• Prove the kernel of a ring homomorphism is an ideal

Properties of Modules

• Show the direct sum and direct product of modules are also modules
• Prove the kernel of a module homomorphism is a submodule
• Demonstrate the image of a module homomorphism is a submodule

Quotient Rings and Modules

Construction and Structure of Quotient Rings

• Given a ring R and an ideal I, the quotient ring R/I is the set of cosets of I in R with addition and multiplication defined on the cosets
• The elements of the quotient ring R/I are of the form a + I, where a ∈ R, and two cosets a + I and b + I are equal if and only if a - b ∈ I
• The First Isomorphism Theorem for rings states if f: R → S is a ring homomorphism, then R/ker(f) ≅ im(f)

Construction and Structure of Quotient Modules

• Given a module M and a submodule N, the quotient module M/N is the set of cosets of N in M with addition defined on the cosets and scalar multiplication inherited from M
• The elements of the quotient module M/N are of the form m + N, where m ∈ M, and two cosets m + N and n + N are equal if and only if m - n ∈ N
• The First Isomorphism Theorem for modules states if f: M → N is a module homomorphism, then M/ker(f) ≅ im(f)

Homomorphisms of Rings and Modules

Ring Homomorphisms

• A ring homomorphism is a function between two rings that preserves the ring operations (addition and multiplication)
• The kernel of a ring homomorphism f: R → S is the set of elements in R that map to the additive identity in S, ker(f) = {r ∈ R | f(r) = 0_S}
• The image of a ring homomorphism f: R → S is the set of elements in S that are mapped to by elements in R, im(f) = {f(r) | r ∈ R}

Module Homomorphisms

• A module homomorphism is a function between two modules that preserves the module operations (addition and scalar multiplication)
• The kernel of a module homomorphism f: M → N is the set of elements in M that map to the additive identity in N, ker(f) = {m ∈ M | f(m) = 0_N}
• The image of a module homomorphism f: M → N is the set of elements in N that are mapped to by elements in M, im(f) = {f(m) | m ∈ M}