Let $B$ be a unital commutative Banach algebra, and let $A$ be an arbitrary subalgebra of $B$. Suppose that an ideal $I\subset A$ consists of elements that are non-invertible in $B$. Then there exists an ideal $J$ in $A$ that: (i) contains $I$, (ii) also consists of elements non-invertible in $B$, and (iii) is of codimension one in $A$. This result is used in the study of a class of subspectra on $A$.