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The kernel of a ring homomorphism is a two-sided ideal
Statement
The kernel of a ring homomorphism is a two-sided ideal.
Facts & Assumptions
Given: A ring homomorphism .
A ring homomorphism preserves addition, multiplication, , and (Ring homomorphism: additive, multiplicative, and required to send to ).
Ring homomorphisms preserve additive inverses (A ring homomorphism satisfies , and for , carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms).
The group kernel is the inverse image of in the additive groups (The kernel and image of a group homomorphism).
A two-sided ideal is an additive subgroup with two-sided absorption (Left, right and two-sided ideals).
Proof
The additive-group kernel of is an additive subgroup of .
If and , then and .
Thus the additive and absorption properties make a two-sided ideal.
Depends on
- Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
- A ring homomorphism satisfies $f(0) = 0$, $f(-a) = -f(a)$ and $f(ma) = m f(a)$ for $m \in \mathbb{Z}$, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms
- The kernel and image of a group homomorphism
- Left, right and two-sided ideals
Used by
- The augmentation map ε:R[G]→ R and the augmentation ideal I_G=kerε Definition
- R×{0} is the kernel of R× S→ S, so (R× S)/(R×{0})≅ S Example
- Commutative rings form a reflective full subcategory of rings Theorem
- Correspondence theorem: ideals of R/I correspond to ideals of R containing I Theorem
- First isomorphism theorem for rings: R/ker f congimf Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Ideals and Quotient Rings (standard reference, not scraped)