Remark: Usually, (left) ideals are defined as a subset of a (unital or non-unital) ring that is a subgroup of the additive group of the ring that "absorbs multiplication from the left by elements of the ring", see Wikipedia https://en.wikipedia.org/wiki/Ideal_(ring_theory) (19.02.2025), or the definition 4 in [BourbakiAlg1] p. 103 and the definition in [Lang] p.86, although a ring is to be considered unital (and commutative!) here, see definition 1 in [BourbakiAlg1] p. 96 resp. the definition in [Lang] p. 83, or definition in [Roman] p. 20.
In contrast, the definition of , does not require the subset to be a subgroup of the additive group, as can be seen by islidl. If is a unital ring, however, it can be proven that each ideal in is a subgroup of the additive group of the ring, see lidlsubg. This is not possible for arbitrary non-unital rings, because the proof uses the existence of the ring unity.