For a given prime p, we study the properties of the p-dissection identities of Ramanujan's theta functions ψ(q) and f (−q), respectively. Then as applications, we find many infinite family of congruences modulo 2 for some ℓ-regular partition functions, especially, for ℓ = 2, 4,5,8,13,16. Moreover, based on the classical congruences for p(n) given by Ramanujan, we obtain many more congruences for some ℓ-regular partition functions.

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In view of the modular equation of fifth order, we give a simple proof of Keith's conjecture which is some infinite families of congruences modulo 3 for the 9-regular partition function. Meanwhile, we derive some new congruences modulo 3 for the 9-regular partition function.

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