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Math Help - clean ring

  1. #1
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    clean ring

    A commutative ring is said to be clean if each of its elements can be written as the sum of a unit and an idempotent. for example, every quasi-local ring is a clean ring.
    If R is clean and it is not a finite product of quasi-local rings, can we conclude that the set of idempotent elements of R is infinite?
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  2. #2
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    by "quasi-local" ring did you mean a ring with only one maximal ideal or with finitely many maximal ideals?
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    I mean a ring with unique maximal ideal(local ring).
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    Quote Originally Posted by xixi View Post
    I mean a ring with unique maximal ideal(local ring).
    ok. the answer is yes. i'll prove that if R is clean and the number of its idempotents is finite, then R is a finite direct product of local rings. the proof is by induction on n, the number of idempotents. suppose n = 2, i.e. the only idempotents of R is 0 and 1. then for any x \in R, either x or 1 - x is a unit because R is clean. now let M be a maximal ideal of R and x \notin M. then Rx + M = R and hence rx + y = 1 - x, for some y \in M and r \in R. thus 1-y=(r+1)x. thus, since y \in M and hence y cannot be a unit, 1-y is a unit and so (r+1)x is a unit. it follows that x is a unit. so we just proved that every element outside M is a unit and that means R is a local ring. this completes the proof of our induction base. now suppose n > 2 and the claim is true for any clean ring with the number of idempotents strictly less than n. choose an idempotent e \in R \setminus \{0,1\}. clearly R = Re \oplus R(1-e) and the number of idempotents of Re and R(1-e) is at most n-1 because 1 \notin Re and 1 \notin R(1-e). so to finish the proof we only need to prove that if R is clean, then Re and R(1-e) are clean too. this is quite easy and i'll prove it for Re. the proof for R(1-e) is similar. first note that the identity element of Re is e. now suppose that x = re is any element of Re. since R is clean, we have r = u + f for some unit u \in R and an idempotent f  \in R. then x=re = ue + fe and it is clear that ue is a unit in Re and fe is an idempotent in Re. the proof is now complete.
    Last edited by NonCommAlg; August 22nd 2011 at 04:50 PM.
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