A Note on Generalizing Almost Clean Rings

A ring 𝑅 is called almost clean when every element of 𝑅 can be written as the some of an idempotent and a regular (nonzero divisor) element. Let 𝑛 ∈ 𝑁. We call a ring n-almost clean if every element of 𝑅 can be written as 𝑎= 𝑒+ 𝑢 1 +𝑢2 +...+𝑢𝑛 where e is an idempotent and every 𝑢𝑖 (1 ≤𝑖≤𝑛) is a regular element and for a fixed polynomial 𝑔(𝑥)∈𝐶(𝑅)[𝑥] where 𝐶(𝑅) denote the cener of R. we say 𝑅 is 𝑔(𝑥)-almost clean when every element of 𝑅 can be written as the sum of an idempotent and a root of 𝑔(𝑥). Finally 𝑅 is said to be (𝑛,𝑔(𝑥))-almost clean when 𝑔(𝑥)∈𝐶(𝑅)[𝑥] and each element 𝑎∈𝑅 can be written as 𝑎= 𝑢 1 + 𝑢2 +...+𝑢𝑛 + 𝑟 where 𝑔(𝑟)= 0 and every 𝑢𝑖 (1 ≤𝑖≤𝑛) is a regular element. we are going to investigate some properties of these rings