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    On the genus and crosscap of the extended sum annihilating-ideal graph of commutative rings
    (Springer, 2025) Nazim, Mohd; Ur Rehman, Nadeem; Abdioğlu, Cihat; Mir, Shabir Ahmad; Nazim
    In the context of a commutative ring with unity, denoted as S, and its associated set of annihilating ideals A(S), there exists a graph known as the extended sum annihilating-ideal graph, denoted as AGΩ(S). This graph has its vertex set derived from the set A(S)*, and it exhibits a specific pattern of connections between its vertices. More precisely, two distinct vertices, referred to as ℑ1 and ℑ2, are linked by an edge if and only if one of the following conditions holds: either ℑ1ℑ2=0 or ℑ1 +ℑ2 ∈ A(S). In the following research paper, we delve into the classification of Artinian commutative rings, denoted as S, with a particular focus on those where the extended sum annihilating-ideal graph takes on one of three distinct forms: a double toroidal graph, a projective plane, or a Klein bottle. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.