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I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition). In the case where $R=k$ is a field, any such algebra decomposes as a finite product of finite separable extensions of $k$.

What about algebras over more general rings? Do we get a decomposition as a finite product of "simpler" rings? For example, what if:

$R$ is local artinian

$R$ is artinian

$R$ is local noetherian of dimension 1

$R$ local noetherian domain of dimension 1

$R$ is a DVR

$R$ is a PID

$R$ is a Dedekind domain

and so on...

I would be interested in any of these cases and in others you may want to add ( I hope the question is not too broad)

In cases where $R$ is normal, can we use Serre's criterion to get a decomposition?

I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition). In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite product of finite separable extensions of $k$.

What about algebras over more general rings? Do we get a decomposition of $A$ as a finite product of "simpler" rings?

For example, what if:

$R$ is local artinian

$R$ is artinian

$R$ is local noetherian of dimension 1

$R$ local noetherian domain of dimension 1

$R$ is a DVR

$R$ is a PID

$R$ is a Dedekind domain

and so on...

I would be interested in any of these cases and in others you may want to add ( I hope the question is not too broad)

In cases where $R$ is normal, can we use Serre's criterion to get a decomposition of $A$?