Finite frames, P-frames and basically disconnected frames

作者：Acharyya Sudip Kumar^*; Bhunia Goutam; Ghosh Partha Pratim

来源：ALGEBRA UNIVERSALIS, 2014, 72(3): 209-224.

DOI：10.1007/s00012-014-0296-x

• P-frame
• basically disconnected frame
• ideal of closed quotients of a frame
• support of real valued continuous functions on a frame
• continuous frame
• P-continuous frame
• upper ideal
• Noetherian ring
• Artinian ring
• coherent ring

Let be an ideal of closed quotients of a completely regular frame L and the collection of all functions in the ring whose support belong to . We show that is a Noetherian ring if and only if is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if is the ideal of all compact closed quotients of L and L is -continuous, then is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of is of the form for some choice of . We furnish equivalent conditions for to be a prime ideal, a free ideal, and an essential ideal of separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if is a coherent ring.

• 出版日期2014-11