This paper is devoted to the study of the inner structure of graded algebras of arbitrary dimension over an arbitrary base field $ \mathbb{F} $. By employing a framework that does not require additional identities such as associativity, commutativity, Lie, or Jordan, we investigate the decomposition of these algebras through connection techniques applied to the support of the grading. We show that, under certain mild conditions on the zero-component, any such algebra can be decomposed into a direct sum of orthogonal graded ideals. Furthermore, we provide conditions for the algebra to be represented as a direct sum of its simple graded ideals. Our results extend several well-known decomposition theorems from the commutative algebra case to the arbitrary algebra setting.
Citation: Antonio J. Calderón. Structural decomposition of arbitrary graded algebras[J]. AIMS Mathematics, 2026, 11(9): 29022-29037. doi: 10.3934/math.20261154
This paper is devoted to the study of the inner structure of graded algebras of arbitrary dimension over an arbitrary base field $ \mathbb{F} $. By employing a framework that does not require additional identities such as associativity, commutativity, Lie, or Jordan, we investigate the decomposition of these algebras through connection techniques applied to the support of the grading. We show that, under certain mild conditions on the zero-component, any such algebra can be decomposed into a direct sum of orthogonal graded ideals. Furthermore, we provide conditions for the algebra to be represented as a direct sum of its simple graded ideals. Our results extend several well-known decomposition theorems from the commutative algebra case to the arbitrary algebra setting.
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Antonio J. Calderón. Structural decomposition of arbitrary graded algebras[J]. AIMS Mathematics, 2026, 11(9): 29022-29037. doi: 10.3934/math.20261154
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