Evaluate an integer polynomial at two. Adding polynomials before evaluating gives the same result as adding their values; multiplying before evaluating also agrees with multiplying their values. The constant polynomial one evaluates to one. This is a unital ring homomorphism.
Now consider multiplying every integer by two. It preserves addition, but φ(ab)=2ab differs from φ(a)φ(b)=4ab. It also sends one to two. An additive-group homomorphism need not preserve the ring structure.
Predict
Which polynomials evaluate to zero at two? Predict that they are exactly the multiples of x−2. Merely listing x−2 and x²−4 supplies examples; proving the entire kernel requires division and its constant remainder.
Experiment
Divide f(x)=x³−3x+2 by x−2. The quotient is x²+2x+1 and the remainder is four. Substitution at two independently gives eight minus six plus two, also four.
Polynomial
Value at two
In the kernel?
x−2
0
Yes
x²−4
0
Yes
x³−3x+2
4
No
Constant c
c
Only when c=0
Over a field F, division by x−a gives f=q(x−a)+r with constant r. Evaluation shows r=f(a). Hence the evaluation kernel is (x−a), and constants prove surjectivity onto F.
Understand
A unital ring homomorphism φ:R→S preserves addition, multiplication and identity. Additive preservation supplies φ(0)=0 and φ(−a)=−φ(a). Its kernel is the set sent to zero. If a,b lie in the kernel, their difference does too. If r∈R and a is in the kernel, φ(ra)=φ(r)φ(a)=0. Thus the kernel is an ideal.
The image is a unital subring, not necessarily an ideal of the codomain. The inclusion Z→Q has image Z, which fails absorption by arbitrary rational scalars. Distinguishing image and kernel properties prevents an unjustified quotient by an image.
The first ring isomorphism theorem states R/ker φ≅im φ. Define the map a+ker φ↦φ(a). Two representatives differ by a kernel element exactly when their images agree, proving well-definedness and injectivity together. Surjectivity onto the image is definitional. The original homomorphism identities give preservation of both operations and of identity. This is more than an additive-group isomorphism.
For evaluation, the theorem gives F[x]/(x−a)≅F. For integer reduction, it gives Z/nZ as the image of the canonical map from Z to residues. More generally, a ring map collapses precisely its kernel cosets; a nontrivial kernel is the obstruction to injectivity.
Our identity-preserving convention excludes the zero map between nonzero rings. That map preserves addition and multiplication but sends one to zero. If the codomain is the zero ring, its zero is also its identity, so the unique map into it is unital. There is no unital map from the zero ring to a nonzero ring, because the same domain element would need to map to both zero and one.
Apply
Audit reduction from Z/18Z to Z/6Z: changing a representative by eighteen does not change its residue modulo six, so the map is well-defined. Find the kernel, image and induced quotient isomorphism. Then ask why the same representative formula from Z/6Z to Z/18Z would fail.
In a proof, declare the codomain before claiming surjectivity. Every map is onto its image, but that image can be a proper subring of the chosen target. Use an explicit preimage construction when onto-ness matters.
Continue
Prime and maximal ideals next let us read domain and field properties directly from a quotient. Properness will be essential because the zero ring is neither under our conventions.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For reduction Z/18Z→Z/6Z, how many elements are in the kernel?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Evaluate f(x)=2x²−3x+1 at x=3 in Z.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove the kernel of a unital ring homomorphism is an ideal.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove R/ker φ is isomorphic as a unital ring to im φ.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For a∈F, prove evaluation F[x]→F has kernel (x−a).
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why the zero map from a nonzero ring to a nonzero ring is excluded by our homomorphism convention.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
