Category:Equivalence of Definitions of Unital Associative Commutative Algebra

This category contains pages concerning Equivalence of Definitions of Unital Associative Commutative Algebra:

Let $A$ be a commutative ring with unity.

Correspondence

Let $B$ be a algebra over $A$ that is unital, associative and commutative.

Let $\struct {C, f}$ be a ring under $A$.

The following statements are equivalent:

$(1): \quad C$ is the underlying ring of $B$ and $f: A \to C$ is the canonical homomorphism to the unital algebra $B$.

$(2): \quad B$ is the algebra defined by $f$.

Homomorphisms

Let $\struct {B, f}$ and $\struct {C, g}$ be rings under $A$.

Let $h: B \to C$ be a mapping.

The following statements are equivalent:

$(1): \quad h$ is a morphism of rings under $A$.

$(2): \quad h$ is a unital algebra homomorphism from the algebra defined by $f$ to the algebra defined by $g$.