Abstract Algebra: Rings & Ideals Quiz

A set with operations * and + such that (R, +) is an abelian group, there exists a 1 ∈ R such that 1 * a = a * 1 = a, a * (b * c) = (a * b) * c, and a * (b + c) = a * b + a * c, (b + c) * a = b * a + c * a.

True

Z, Z ln Z, IR, R[x], Q.

<p>An element a ≠ 0 in R such that ab = 0 for some b ≠ 0 in R.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>Q, IR, Z/pZ for prime p.</p> Signup and view all the answers

<p>A set S ⊆ R such that 1 ∈ S, (S, +) is a subgroup of (R, +), and a, b ∈ S implies ab ∈ S.</p> Signup and view all the answers

<p>A set I ⊆ R such that (I, +) is a subgroup of (R, +) and for x ∈ I, r ∈ R, it holds that r * x ∈ I and x * r ∈ I.</p> Signup and view all the answers

<p>nZ' ⊆ Z', principal ideal = {rx : r ∈ R} in a commutative ring R.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>pZ ∈ Z for prime p, ⊆ F[x] for g(x) irreducible.</p> Signup and view all the answers

<p>A map f: R → S such that f(1) = 1, f(a + b) = f(a) + f(b), and f(ab) = f(a) f(b).</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>Ker(f) = {a ∈ R : f(a) = 0}.</p> Signup and view all the answers

<p>The smallest n such that 1 + 1 + ... + 1 = 0 (n times).</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>R/I = {I + a : a ∈ R} is a ring with structure defined through operations.</p> Signup and view all the answers

<p>If f: R → S is a surjective homomorphism, then R/Ker(f) ≅ S.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>Z → R with Kernel char(R) Z.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>f(x) = a₀ + ... + aₙ X^n ∈ Z[X] implies s I a₀ and t | aₙ for the root to exist.</p> Signup and view all the answers

<p>f(x) = a₀ + ... + aₙ X^n ∈ Z[X] has prime p such that p | aᵢ for 0 ≤ i ≤ n-1, pX aₙ, and p² | a₀ indicates that f(x) is irreducible in Q[x].</p> Signup and view all the answers

<p>A set G with operation * such that a * (b * c) = (a * b) * c, e ∈ G such that e * g = g * e = g, and for any g ∈ G, g⁻¹ ∈ G such that g * g⁻¹ = g⁻¹ * g = e.</p> Signup and view all the answers

<p>Z, Z^1 / nZ, S_n, D_n, A_n.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>A subset H of G such that H ≠ ∅, a, b ∈ H implies ab ∈ H, and a ∈ H implies a⁻¹ ∈ H.</p> Signup and view all the answers

<p>The cyclic subgroup {gⁿ : n ∈ Z} generated by g ∈ G, A_n ⊆ S_n.</p> Signup and view all the answers

<p>The smallest n ≥ 1 such that g^n = e.</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>A map f: G → H of groups such that f(ab) = f(a) f(b).</p> Signup and view all the answers

<p>True</p> Signup and view all the answers

<p>S_3 ≅ D_3, Z / 2Z * Z / 3Z ≅ Z / 6Z.</p> Signup and view all the answers

<p>Ker(f) = {g ∈ G : f(g) = e}.</p> Signup and view all the answers