Abstract Algebra

Which of the following condition(s) needs to be satisfied in order for a set to be a ring?

• Abelian group under addition
• monoid under multiplication
• multiplication is distributive over addition
• all of these needs to be satisfied

Being classified as abelian group in the concept of rings means

• (a + b) + c = a + (b + c) for all a, b, c in R
• a + b = b + a for all a, b in R
• Option 3
• all of these

In abstract algebra, a monoid is defined as

• a set equipped with a binary operation and identity element
• a set equipped with a binary operation only
• a set equipped with an identity element and its inverse
• a set equipped with abelian properties

The expression Z/4Z qualifies to be a ring.

• true
• false
• neither
• can not be determined, not enough data was provided

The expression below qualifies as a ring

• true
• false
• neither
• can not be determined, not enough data was provided

He defined the concept of the ring of integers of a number field

• Dedekind
• Hilbert
• Fraenkel
• Noether

A function f from R to S that preserves the ring operations is called

• Ring homomorphism
• Ring isomorphism
• Multiplicative Ring
• Abelian Ring

If there exists an inverse homomorphism to f, then f is said to be

• isomorphism
• abelian
• multiplicative
• monoid

A commutative ring is defined as

• a ring in which the multiplication operation is commutative
• a ring in which the addition operation is commutative
• a ring in which it is homomorphic
• a ring in which it is isomorphic

A subring A ≤ R is called a(n) _______ of R if ∀r ∈ R and ∀a ∈ A we have ar, ra ∈ A.

• ideal of R
• isomorphic to R
• homomorphic to R
• subset to R