Characterizations On Almost Prüfer Domains

In this paper, the basic characters and theories of almost Prüfer domainsare studied. Firstly, we discuss the ideals and extension rings of almost Prüferdomains. It is showed that R is an almost Prüfer domain if and only if forA,B,C∈S, A (B + C) = (A B) + (A C), if and only if for A,B,C∈S,A(B C) = AB AC, if and only if for A,B∈S, (A + B)(A B) = AB,where S ={A | A is an ideal of R and for some positive integer n, {ai} (?) R-{0},i = 1,2,···, A = ({a_iⁿ })}. We also prove that if R is an almost Prüfer domain,T is an overring of R, M is a nonzero prime ideal of T, then RM TR (?) TM is aroot extension. Secondly, the dimensions and fractional rings of polynomial ringsabout almost Prüfer domains are characterized. It is showed that if R is an almostPrüfer domain, then dimR[X₁,···,X_n] = dimR + n. and let R be a domain, ifR X (?) R^c X is a root extension, then R is an almost Prüfer domain if and onlyif R X is an almost Prüfer domain. Besides, the relationships between almostPrüfer domains and several other domains are investigated. Finally, the inverselimits of almost Prüfer domains are studied. It is proved that the inverse limit ofalmost valuation domain is an almost valuation domain, and the inverse limit ofalmost Prüfer domain is an almost Prüfer domain under the riding assumption.In general cates, an example of the inverse limit of almost Prüfer domain that isnot a almost Prüfer domain is presented.