Photorefractive keratectomy with intraoperative application of MMC was a safe procedure that produced excellent visual outcomes with few complications.
We first construct an NI ring but not 2-primal from given any 2-primal ring, in a simpler way than well-known examples. We study the structure of NI rings relating to strongly prime ideals and show that minimal strongly prime ideals can be lifted in NI rings. A ring is called (respectively weakly) pm if every (respectively strongly) prime ideal is contained in a unique maximal ideal in it. For a 2-primal ring R Sun proved that R is pm if and only if Max(R) is a retract of Spec(R) if and only if Spec(R) is normal. In the present note we prove for an NI ring R that R is weakly pm if and only if Max(R) is a retract of SSpec(R) if and only if SSpec(R) is normal, where SSpec(R) is the space of strongly prime ideals of R. We also prove that R is weakly pm if and only if R is pm when R is a symmetric ring. We lastly consider several kinds of extensions of NI rings.
Abstract. We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.
Abstract. We study the structure of the set of nilpotent elements in various kinds of ring and introduce the concept of NR ring as a generalization of Armendariz rings and N I rings. We determine the precise relationships between N R rings and related ring-theoretic conditions. The Köthe's conjecture is true for the class of N R rings. We examined whether several kinds of extensions preserve the N R condition. The classical right quotient ring of an N R ring is also studied under some conditions on the subset of nilpotent elements.
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