Does there exist any term (or, maybe, a "description"?) for commutative unital noetherian rings such that their Jacobson ideals are prime (and so, their maximal spectra are irreducible)? What is the relation of this condition to the primality of the nilradical?
Upd. Sorry; I was really stupid! Actually, the primality of the Jacobson radical is "independent from" that of the nilradical. Indeed, there exist local rings whose nil radical is not prime (see Konstantin Ardakov's example); on the other hand, a "semi-localization" of a indecomposable regular ring at any two closed points (take $R=k[X]$ and invert all polynomials whose values in $0$ and $1$ are non-zero) is regular indecomposable but its maximal spectrum consists of two closed points!
So, I am rather interested in the following question: for which $R$ its Jacobson radical contains a prime ideal? My collegues has told me that this is equivalent to the fact that the closed points of $Spec R$ lie in some single irreducible component of it, and there is a paper https://www.math.hmc.edu/~henriksen/publications/1977_Henriksen_Some_sufficient_conditions_for_the_Jacobson_radical_of_a_commutative_ring_with_identity_to_contian_a_prime_ideal.pdf on this question. Yet any further comments would be very welcome!