Modules and Lattice Theory Self Test

Ring and Lattice Theory Quiz

Test your knowledge of fundamental definitions and concepts

Q1

A Noetherian ring is defined as a ring that satisfies:

The ascending chain condition on ideals The descending chain condition on ideals That every element is a unit That it has no zero divisors

Q2

An Artinian ring is defined as a ring that satisfies:

The ascending chain condition on ideals The descending chain condition on ideals That every element is nilpotent That it is a field

Q3

Hilbert's Basis Theorem states that:

If R is a Noetherian ring, then R[x] is also Noetherian If R is a Noetherian ring, then R[x] is Artinian If R is an Artinian ring, then R[x] is also Artinian If R is a field, then R[x] is a field

Q4

Krull's Intersection Theorem relates to:

The intersection of powers of an ideal in a Noetherian ring The intersection of all prime ideals in a ring The intersection of all maximal ideals in a ring The intersection of all ideals in a ring

Q5

A prime ideal P in a commutative ring R is defined by the property:

If ab ∈ P, then a ∈ P or b ∈ P If a + b ∈ P, then a ∈ P or b ∈ P If a ∈ P, then a² ∈ P If a ∈ P, then -a ∈ P

Q6

A primary ideal Q in a commutative ring R is defined by the property:

If ab ∈ Q and a ∉ Q, then bⁿ ∈ Q for some positive integer n If ab ∈ Q, then a ∈ Q or b ∈ Q If a ∈ Q, then a² ∈ Q If a ∈ Q, then -a ∈ Q

Q7

A partially ordered set (poset) is a set equipped with a binary relation that is:

Reflexive, antisymmetric, and transitive Reflexive, symmetric, and transitive Irreflexive, antisymmetric, and transitive Symmetric and transitive

Q8

A lattice is a partially ordered set in which:

Every pair of elements has a least upper bound and a greatest lower bound Every pair of elements is comparable There is a unique minimal element There is a unique maximal element

Q9

A modular lattice satisfies the modular law:

If x ≤ z, then x ∨ (y ∧ z) = (x ∨ y) ∧ z x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) If x ≤ z, then x ∧ (y ∨ z) = (x ∧ y) ∨ z

Q10

A complemented modular lattice is a modular lattice in which:

Every element has a complement Every element is its own complement There is a unique complement for each element The lattice has a unique minimal element

Q11

A sublattice of a lattice L is:

A subset of L that is closed under the meet and join operations of L Any subset of L A subset of L that is closed under only the meet operation A subset of L that is closed under only the join operation

Q12

An ideal in a lattice L is a subset I of L such that:

If x ∈ I and y ≤ x, then y ∈ I; and if x, y ∈ I, then x ∨ y ∈ I If x ∈ I and y ≥ x, then y ∈ I; and if x, y ∈ I, then x ∧ y ∈ I If x, y ∈ I, then x ∨ y ∈ I and x ∧ y ∈ I If x ∈ I and y ≤ x, then y ∈ I; and if x, y ∈ I, then x ∧ y ∈ I

Q13

A lattice homomorphism between two lattices L and M is a function f: L → M that:

Preserves meets and joins: f(x ∧ y) = f(x) ∧ f(y) and f(x ∨ y) = f(x) ∨ f(y) Preserves only meets: f(x ∧ y) = f(x) ∧ f(y) Preserves only joins: f(x ∨ y) = f(x) ∨ f(y) Preserves order: if x ≤ y in L, then f(x) ≤ f(y) in M

Q14

The ascending chain condition (ACC) for ideals in a ring R states that:

Every ascending chain of ideals in R stabilizes after a finite number of steps Every descending chain of ideals in R stabilizes after a finite number of steps Every ideal in R is finitely generated Every prime ideal in R is maximal

Q15

The descending chain condition (DCC) for ideals in a ring R states that:

Every descending chain of ideals in R stabilizes after a finite number of steps Every ascending chain of ideals in R stabilizes after a finite number of steps Every ideal in R is finitely generated Every prime ideal in R is maximal

Quiz Results

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