For a commutative Artinian unital ring, it is well known that every ideal contains at least a minimal ideal, a non-zero ideal that dose not contain a proper non-zero ideal. In general, not Artinian case, are rings with above property important or are they a famous class of rings, or are there any characterization or description for them?
If every nonzero right ideal of a ring $R$ contains a minimal right ideal, you usually express this by saying that $R$ has an essential right socle.
This expression is frequently used in the literature: try for example this query in google books.
I am not aware of alternative characterizations of this, although it is used in conjunction with other conditions to make characterizations. I think I saw it frequently used in Faith's work on pseudo-Frobenius and finitely-pseudo-Frobenius rings.
If $A$ is a noetherian commutative domain and is not a field, then $A$ has no minimal non-zero ideal at all.
Let $\mathfrak a$ be a minimal ideal. We have $\mathfrak a^2=\mathfrak a$ and thus by NAKAYAMA there exists $e\in \mathfrak a$ with $a=ea$ for all $a\in \mathfrak a$. But then $e=e^2$, hence $e=0$ or $e=1$ so that $\mathfrak a=0$ or $\mathfrak a=A$.
Given an integral noetherian affine scheme $X$, no closed subscheme $\emptyset \subsetneq Y\subsetneq X$ is maximal .