in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

This study investigates the causal relationship between clean and non-clean energy consumption and economic growth in Brazil over the period of 1980–2009. Clean energy consumption at aggregated level of total renewable energy consumption and disaggregated levels of hydroelectric, new renewables, and nuclear energy consumption are tested within a production function framework. A cointegration te...