41. The converse does not hold : every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain . 42. An integral domain is completely integrally closed if and only if the monoid of divisors of " A " is a group. 43. An element of an integral domain is prime if, and only if, an ideal generated by it is a nonzero prime ideal. 44. It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains . 45. *PM : polynomial ring over integral domain , id = 6918-- WP guess : polynomial ring over integral domain-- Status: 46. *PM : polynomial ring over integral domain, id = 6918-- WP guess : polynomial ring over integral domain -- Status: 47. One would think that integral domains and fields should be given more prominence in the article and should be discussed earlier in and more depth. 48. If " A " is an integral domain , then " d " is the transcendence degree of its field of fractions. 49. In general, suppose R is an integral domain and f is a monic univariate polynomial of degree d \ geq 1 with coefficients in R. 50. A commutative principal ideal ring which is also an integral domain is said to be a " principal ideal domain " ( PID ).
