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We develop a new approach to RG flows and show that one-loop flows inmultiscalar theories can be described by commutative but non-associativeEuclidean algebras. As an example related to $D$-brane field theories andtensor models, we study the algebra of a theory with $M$ $SU(N)$ adjointscalars and its large $N$ limits. A block diagonal decomposition of the RG flowinto independent sets of couplings is achieved by a complete decomposition ofthe algebra into a sum of simple ideals, permitted by its Euclidean metric. Thealgebraic concepts of idempotents and Peirce numbers/Kowalevski exponents areused to further characterise the RG flows.  We classify and describe all large N limits of algebras of multiadjointscalar models: the standard `t Hooft matrix theory limit, a `multi-matrix'limit, each with one free parameter, and an intermediate case with extrasymmetry and no free parameter of the algebra, but an emergent free parameterfrom a line of one-loop fixed points. The algebra identifies these limitswithout diagrammatic or combinatorial analysis.

Classifying large N limits of multiscalar theories by algebra