If the set S = { diag(a, b) : a, b ∈ ℤ } (the 2 × 2 diagonal integer matrices)…

A left ideal of a ring R is a subring I such that r·x ∈ I for every r ∈ R and every x ∈ I — I must absorb multiplication by the whole ring from the left. A right ideal requires the symmetric condition x·r ∈ I for every r ∈ R and x ∈ I — absorption from the right. A subring can satisfy neither, one, or both conditions, and the only way to settle it is to test the ring’s actual multiplication rule directly.

Here R = M₂(ℤ) and S = { diag(a, b) : a, b ∈ ℤ } is the subring of diagonal integer matrices. Because absorption must hold for every matrix in R, a single matrix M for which the product leaves S is enough to rule out that side’s closure.

1. Left absorption: take A = [[1, 0], [0, 0]] ∈ S and M = [[0, 0], [1, 0]] ∈ M₂(ℤ). Then M·A = [[0, 0], [1, 0]], which is not diagonal, so M·A ∉ S — left absorption fails for this M.

2. Right absorption: take the same A = [[1, 0], [0, 0]] ∈ S and M = [[0, 1], [0, 0]] ∈ M₂(ℤ). Then A·M = [[0, 1], [0, 0]], which is not diagonal, so A·M ∉ S — right absorption fails for this M.

Cross-check: both counterexamples use a matrix with a single off-diagonal 1 multiplying a diagonal matrix; either side of that product mixes a row or column and produces an off-diagonal entry, so the failure is structural rather than a coincidence of the specific numbers chosen — any diagonal entry a ≠ 0 in A gives the same kind of counterexample on both sides.

Since a matrix M exists that breaks left absorption and a matrix M exists that breaks right absorption, S satisfies neither the left-ideal condition nor the right-ideal condition, even though it is a subring of M₂(ℤ).