Equivalent definitions of a local algebra

Question

Let $A$ be a unital, associative commutative algebra over a field $k$. In particular, $A$ is a ring and we call $A$ local if it is local as a ring. Namely, call $A$ local if $A$ has a unique maximal ideal.

I have encountered the following different definition of a local algebra: $A$ is called local if it is finite-dimensional as a $k$-algebra and if $A/J(A)\cong k$. Here $J(A)$ denotes the Jacobson radical of $A$.

Are these two definitions equivalent?

Answer 1

Using these definitions given

1. There exists an algebra satisfying the first definition, but failing to satisfy the finite dimensionality criterion of the second: $A=\mathbb R$ and $k=\mathbb Q$.

2. There exists an algebra satisfying the first definition, and the finite dimensionality condition, but not the isomorphism in the second condition: $A=\mathbb C$ and $k=\mathbb R$.

These can be tweaked, of course, so that $A$ isn't just a field.

What's true in general is that (for a commutative ring) $R/J(R)$ is a field iff $R$ has a unique maximal ideal. But if $R$ is a $k$ algebra, the field $R/J(R)$ can be quite different from $k$. So, you can see the second definition is defining something much more special than an algebra that's simply a local ring.