We define the concept of $I$-stable ideals in the ring of commutative polynomials over a field, generalizing the so-called stable ideals, which arise as ideals of higher terms under general linear changes of variables. The interest in ideals of this type is motivated by the fact that certain problems concerning homogeneous ideals (for example, the problem of obtaining upper estimates for the graded Betti numbers) can be reduced to the study of stable ideals. $I$-stable ideals retain many interesting properties of stable ideals. In particular, the minimal resolutions of $I$-stable ideals constructed in this paper enable us to obtain an explicit formula for the graded Betti numbers, which turn out to be independent of the characteristic of the ground field. Factor rings by $I$-stable ideals generated by monomials of degree $\geqslant 2$ are Golod rings. We also consider other analogues of stable ideals (strongly and weakly $I$-stable ideals) and give conditions sufficient for the factor ring by an $I$-stable ideal to be Cohen–Macaulay or Gorenstein.