It is true that an ideal of semisimple ring is semisimple ring. It is true that an ideal of an Artinian semisimple ring is an Artinian ring. What about just Artinian? Is there an example with the following conditions:
A ring $R$ is Artinian.
$K$ is an ideal in $R$.
$K$ is not Artinian, i.e. we have ideals $A_i\unlhd K$, so that $$K\supsetneq A_1\supsetneq A_2\supsetneq A_3\supsetneq\dots$$
It seems not to be hard, but i can't find enough good examples of Artinian non-semisimple rings. It does not matter if we talk about commutative or just associative rings.
Consider the ring $R$ that is the trivial extension $\mathbb Q\ltimes\mathbb Q$, that is, the set $\mathbb Q\times \mathbb Q$ with coordinatewise addition and with multiplication given by $(a,b)(c,d)=(ac, ad+bc)$.
It's easy to see that $R$ is a $2$ dimensional $\mathbb Q$ algebra, so it is certainly Artinian. It's also clearly commutative, and has identity $(1,0)$.
But the ideal $I=\{0\}\times \mathbb Q$ squares to zero, so it is a rng whose product is zero. In such a rng, every additive subgroup is an ideal, and $\mathbb Q$ has infinitely ascending and descending subgroups, so it has neither the ACC nor DCC on ideals.
Another good example, whose explanation is very much the same, would be the $2\times 2$ upper triangular matrices over $\mathbb Q$ (but it is not commutative.). It's $3$ dimensional, but has an ideal squaring to zero.