Ideals - Special Subrings

Abstract Algebra · Axiom Academy

LESSON Ideals: Special Subrings Understanding the absorption property and why ideals are the natural kernels of ring homomorphisms I is a subgroup under addition (closed, has 0, has inverses) Absorption: For all r ∈ R and i ∈ I , we have ri ∈ I and ir ∈ I The absorption property is what distinguishes ideals from arbitrary subrings. When you multiply an ideal element by any ring element (from either side), the result stays in the ideal. Depending on which absorption condition we require, we get different types of ideals: In commutative rings, all three concepts coincide since ri = ir . But in non-commutative rings (like matrix rings), the distinction matters greatly. One of the most beautiful results in ring theory: ideals are precisely the kernels of ring homomorphisms . This connection explains why ideals are so fundamental. Then ker( φ ) = r ∈ R : φ(r) = 0 is an ideal of R Conversely, every ideal I of R is the kernel of the quotient map R → R/I φ(rx) = φ(r)φ(x) = φ(r) · 0 = 0 In commutative rings, the simplest ideals are those generated by a single element. In ℤ, the ideal ⟨ n ⟩ = n ℤ consists of all multiples of n In ℤ[ x ], the ideal ⟨ x ⟩ consists of all polynomials with zero constant term In ℝ[ x ], the ideal ⟨ x² + 1 ⟩ consists of all multiples of x² + 1

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