Trace Properties and the Rings R(X) and R{x}

An integral domain R has the radical trace property (or is an RTP domain, resp. an LTP domain), if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp.I(R : I)RP = PRP for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this project, we will study the descent of the trace properties from special overrings of and integral domain R to R itself. Also we will investigate the transfer of the trace properties to the Nagata ring R(X) and Serres conjecture ring R{x}i in different contexts of integral domains such as integrally closed domains, Noetherian and Mori domains, pseudo-valuation domains and more.