Questions: Subrings and Ideals

Question 1

Ring R = ℤ and subset I = 2ℤ (all even integers). A student claims 2ℤ is a subring but not an ideal because it doesn't contain the multiplicative identity 1. What is wrong with this reasoning?

• A. The claim is correct — 2ℤ is a subring but not an ideal because ideals must contain 1
• B. 2ℤ is an ideal (it absorbs multiplication from outside ℤ) but not a subring, because it lacks the identity
• C. 2ℤ is an ideal — ideals do not need to contain the multiplicative identity, only to absorb ring multiplication
• D. 2ℤ is neither a subring nor an ideal because it is not closed under multiplication

Answer: (C)

Ideals do not require the multiplicative identity. The definition of an ideal I requires: (1) I is an additive subgroup, and (2) for all r ∈ R and a ∈ I, both ra and ar are in I. For I = 2ℤ and any integer r, the product r·(2k) = 2(rk) is even — so 2ℤ absorbs multiplication from all of ℤ. 2ℤ is also an additive subgroup. Therefore 2ℤ is an ideal. It is NOT a subring under the convention requiring the multiplicative identity (since 1 ∉ 2ℤ), but it is perfectly valid as an ideal.

Question 2

Why can't you form a quotient ring R/S using a subring S the way you can form a quotient group using a normal subgroup?

• A. You can — any subring produces a valid quotient ring
• B. Coset multiplication (r + S)(s + S) = rs + S is not well-defined unless S absorbs multiplication from all of R
• C. Subring cosets don't partition R properly, so the quotient set isn't well-defined
• D. Quotient rings require the subring to be commutative, but not all subrings are

Answer: (B)

For coset multiplication to be well-defined, we need: if r ≡ r' (mod S) and s ≡ s' (mod S), then rs ≡ r's' (mod S). This requires that for any a ∈ S and r ∈ R, the product ra ∈ S — the absorption property of ideals. A subring S is closed under multiplication of its own elements, but rs (for r ∉ S, a ∈ S) need not land in S. Without the absorption property, different representatives of the same coset give different coset products, and the ring structure on R/S collapses. Ideals are exactly the substructure that makes quotient ring multiplication well-defined.

Question 3

The kernel of any ring homomorphism is always a two-sided ideal.

• T. True
• F. False

Answer: True

If φ: R → S is a ring homomorphism and a ∈ ker(φ), then for any r ∈ R: φ(ra) = φ(r)φ(a) = φ(r)·0 = 0, so ra ∈ ker(φ). Similarly ar ∈ ker(φ). The kernel is also an additive subgroup (homomorphisms preserve subtraction). So ker(φ) satisfies all the conditions of a two-sided ideal. Conversely, for every ideal I ⊆ R, the quotient map R → R/I is a ring homomorphism with kernel I. This bijection between ideals and kernels is the central fact of ring theory.

Question 4

Most ideal in a ring is also a subring of that ring.

• T. True
• F. False

Answer: False

This depends on the definition of subring. If subrings are required to contain the multiplicative identity 1 (the most common modern convention), then an ideal I ≠ R typically does not contain 1 — since if 1 ∈ I, then r·1 = r ∈ I for all r ∈ R, forcing I = R. So a proper ideal fails the subring test under this convention. Even under the weaker convention that subrings need not contain 1, an ideal is also a subring, but the more important point is that ideals and subrings serve different structural roles: ideals are the kernels of homomorphisms and the right notion for quotient constructions; subrings are rings in their own right but cannot generally support quotient structures.

Question 5

Why is the absorption property of an ideal — that ra ∈ I and ar ∈ I for all r ∈ R — exactly the right condition to make the quotient R/I a well-defined ring, and why does the weaker closure condition of a subring fall short?

Model answer: The absorption property guarantees that coset multiplication is independent of which representative you choose. If a ∈ I and r ∈ R, then (r + a)s = rs + as, and as ∈ I (by absorption), so rs + as lands in the same coset as rs. Without absorption, choosing a different representative of a coset (by adding an element of I) could shift the product to a different coset, making multiplication ill-defined. A subring S is only closed under multiplication of two elements already in S; it says nothing about r·a when r ∉ S. This means the coset product (r + S)(s + S) = rs + S can be ambiguous, because adding an element of S to r before multiplying by s produces rs + (element of S)·s, and that element of S times s need not stay in S.

The algebraic moral: quotient constructions always require the substructure to 'absorb' the action of the ambient structure. In groups, normal subgroups absorb conjugation (gNg⁻¹ ⊆ N), enabling well-defined quotient group multiplication. In rings, ideals absorb multiplication from outside (rI ⊆ I), enabling well-defined quotient ring multiplication. The ring analogy of 'normal subgroup' is precisely 'ideal' — not 'subring.'