A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of R R is also a direct summand. The ring R is called a right C3 ring if any sum of two independent summands of R R is also a direct summand. It is well known that a right C2 ring must be a right C3 ring but the converse assertion is not true. The ring R is called J -regular if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. Let ℕ be the set of natural numbers and let Λ be any infinite set. The following assertions are proved to be equivalent for a ring R: (1) ℂ\( \mathbb{F}\mathbb{M} \) ℕ R) is a right C2 ring; (2) ℂ\( \mathbb{F}\mathbb{M} \) Λ(R) is a right C2 ring; (3) ℂ\( \mathbb{F}\mathbb{M} \) ℕ(R) is a right C3 ring; (4) ℂ\( \mathbb{F}\mathbb{M} \) Λ(R) is a right C3 ring; (5) ℂ\( \mathbb{F}\mathbb{M} \) ℕ(R) is a J -regular ring and \( \mathbb{M} \) n (R) is a right C2 (or right C3) ring for all integers n ≥ 1.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 12, pp. 1718–1722, December, 2014.
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Shen, L., Chen, J. C2 Property of Column Finite Matrix Rings. Ukr Math J 66, 1933–1938 (2015). https://doi.org/10.1007/s11253-015-1060-x
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DOI: https://doi.org/10.1007/s11253-015-1060-x