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The canonical projection is a surjective ring homomorphism with kernel
Statement
The canonical projection is a surjective ring homomorphism with kernel .
For , has these properties.
Facts & Assumptions
Given: A ring and a two-sided ideal .
is a ring with the stated coset operations (For a two-sided ideal , the additive cosets form a ring with identity ).
A ring homomorphism preserves addition, multiplication, and identity (Ring homomorphism: additive, multiplicative, and required to send to ).
The additive quotient map is surjective (The canonical projection , , is a surjective group homomorphism).
A kernel is the inverse image of the identity element (The kernel and image of a group homomorphism).
The coset-equality criterion gives exactly when , hence exactly when because is an additive subgroup ( iff , and iff ).
Proof
The identities , , and show that is a ring homomorphism.
By [L3] it is surjective; by [L4] and [L5], its kernel is exactly .
Thus .
Depends on
- For a two-sided ideal $I$, the additive cosets form a ring $R/I$ with identity $1+I$
- Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
- The canonical projection $\pi:G\to G/N$, $\pi(g)=gN$, is a surjective group homomorphism
- The kernel and image of a group homomorphism
- $x\in aH$ iff $a^{-1}x\in H$, and $aH=bH$ iff $a^{-1}b\in H$
Used by
- If I⊆ J are ideals of R, then J/I is an ideal of R/I Lemma
- The product of primitive integer polynomials is primitive, and contents multiply Lemma
- For every n∈ℕ, the congruence-class ring ℤ/n is the quotient ring ℤ/nℤ Proposition
- A nonzero polynomial over a field is separable exactly when its gcd with its derivative is 1 Theorem
- A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring Theorem
- Correspondence theorem: ideals of R/I correspond to ideals of R containing I Theorem
- Irreducibility after reduction modulo a prime implies irreducibility over ℚ when the leading coefficient survives Theorem
- R/P is an integral domain if and only if P is a prime ideal Theorem
- Second isomorphism theorem for rings: S/(S∩ I)≅(S+I)/I Theorem
- Third isomorphism theorem for rings: (R/I)/(J/I)≅ R/J Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 29 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Ideals and Quotient Rings (standard reference, not scraped)