It follows that p is irreducible. Clearly p is the generic point of W . Then p contains a or each a i . This proves the assertion. A category is spectral if any non-terminal object has a prime ideal.
Any spectral category is radical.
Since each non-terminal object has a prime ideal which is radical and proper, the category is radical. Suppose C is a spectral category. An ideal of an object in a spectral category is radical if and only if it is spatial. Suppose a is a proper radical ideal of X . This shows that a is an intersection of prime ideals.