The proper terminology is anon-trivial proper ideal

Anyway,

The ring is a ring with unity whose addition operation is matrix addition and multiplication is matrix multiplication.

If is a matrix thatisinvertible then, the ideal containing as an element must contain all elements of (because any ideal containing a unit is improper) thus it is the ring itself.

Thus, the only way a non-trivial ideal can exist in is when all the elements are non-units, that is non-invertible matrices. But that set cannot be an additive subgroup of the ring because addition would not be closed (since the sum of two non-invertible matrices can be invertible).