Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$, namely, $A^e = A \otimes_k A^{op}$.

It seems that there are two typical definitions for separable $k$-algebras. Consider the following two conditions.

1. $A$ is projective as an $A^e$-module.
2. For any field extension $K$ of $k$, the algebra $A \otimes_k K$ is semsimple.

Also consider the following condition.

3. $A$ is finite-dimensional over $k$ and the condition 2 is satisfied.

In some literature (e.g. Corollary 10.6 in Pierce's Associative Algebra), it is shown that 1 is equivalent to 3.

I wonder whether 2 is actually equivalent to 1 and 3, that is,

Question: Suppose that the condition 2 is satisfied. Then is $A$ finite-dimensional over $k$?

I found some people say that this question is true, and even adopt the condition 2 as the definition of separable algebras: e.g. nLab's article on separable algebra, and Lemma 3.3 in Reyes-Rogalski's Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular. I looked up some cited references, but it seems that in the proofs, $A$ is assumed to be finite-dimensional.

I'm not sure whether the question is true even when $A$ is commutative, or $A$ is a filed.

Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$, namely, $A^e = A \otimes_k A^{op}$.

It seems that there are two typical definitions for separable $k$-algebras. Consider the following two conditions.

1. $A$ is projective as an $A^e$-module.
2. For any field extension $K$ of $k$, the algebra $A \otimes_k K$ is semisimple.

Also consider the following condition.

3. $A$ is finite-dimensional over $k$ and the condition 2 is satisfied.

In some literature (e.g. Corollary 10.6 in Pierce's Associative Algebra), it is shown that 1 is equivalent to 3.

I wonder whether 2 is actually equivalent to 1 and 3, that is,

Question: Suppose that the condition 2 is satisfied. Then is $A$ finite-dimensional over $k$?

I found some people say that this question is true, and even adopt the condition 2 as the definition of separable algebras: e.g. nLab's article on separable algebra, and Lemma 3.3 in Reyes-Rogalski's Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular. I looked up some cited references, but it seems that in the proofs, $A$ is assumed to be finite-dimensional.

I'm not sure whether the question is true even when $A$ is commutative, or $A$ is a filed.

Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$.

It seems that there are two typical definitions for separable $k$-algebras. Consider the following two conditions.

1. $A$ is projective as an $A^e$-module.
2. For any field extension $K$ of $k$, the algebra $A \otimes_k K$ is semsimple.

Also consider the following condition.

3. $A$ is finite-dimensional over $k$ and the condition 2 is satisfied.

In some literature (e.g. Corollary 10.6 in Pierce's Associative Algebra), it is shown that 1 is equivalent to 3.

I wonder whether 2 is actually equivalent to 1 and 3, that is,

Question: Suppose that the condition 2 is satisfied. Then is $A$ finite-dimensional over $k$?

I found some people say that this question is true, and even adopt the condition 2 as the definition of separable algebras: e.g. nLab's article on separable algebra, and Lemma 3.3 in Reyes-Rogalski's Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular. I looked up some cited references, but it seems that in the proofs, $A$ is assumed to be finite-dimensional.

I'm not sure whether the question is true even when $A$ is commutative, or $A$ is a filed.

Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$, namely, $A^e = A \otimes_k A^{op}$.

It seems that there are two typical definitions for separable $k$-algebras. Consider the following two conditions.

1. $A$ is projective as an $A^e$-module.
2. For any field extension $K$ of $k$, the algebra $A \otimes_k K$ is semsimple.

Also consider the following condition.

3. $A$ is finite-dimensional over $k$ and the condition 2 is satisfied.

In some literature (e.g. Corollary 10.6 in Pierce's Associative Algebra), it is shown that 1 is equivalent to 3.

I wonder whether 2 is actually equivalent to 1 and 3, that is,

Question: Suppose that the condition 2 is satisfied. Then is $A$ finite-dimensional over $k$?

I found some people say that this question is true, and even adopt the condition 2 as the definition of separable algebras: e.g. nLab's article on separable algebra, and Lemma 3.3 in Reyes-Rogalski's Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular. I looked up some cited references, but it seems that in the proofs, $A$ is assumed to be finite-dimensional.

I'm not sure whether the question is true even when $A$ is commutative, or $A$ is a filed.