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Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. Must $A$ be isomorphic to a subring of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. EDIT: Assume also that $k \longrightarrow A$ is injective. Then: Must $A$ be isomorphic to a $k$-subalgebra of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. Must $A$ be isomorphic to a subring of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

EDIT: More fundamentally, I'm looking for necessary and sufficient conditions on $A$ so that the multiplication map is an isomorphism.

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. Must $A$ be isomorphic to a subring of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. Must $A$ be isomorphic to a subring of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism. Must $A$ be isomorphic to a subring of $K$? If necessary, assume also that $k$ is an integral domain and/or is of characteristic zero.

Conversely: For which $k$-algebras $A$ contained in $K$ is the multiplication map $A \otimes_k A \longrightarrow A$ an isomorphism? Equivalently, for which $k$-algebras $A$ contained in $K$ is $A \otimes_k A$ $k$-torsion-free?

EDIT: More fundamentally, I'm looking for necessary and sufficient conditions on $A$ so that the multiplication map is an isomorphism.