In a group, normal subgroups are most likely not cyclic and hence not generated by a single element. Ideals, as kernels of homomorphisms, is a similar concept to normal subgroups. Why would we expect some integral domains to have the property which all ideals are principle (generated by one single element)?

As a second part of the question, what is the best way to visualize principle ideal domains in general? Since prime ideals are maximal, these ideals "cover a lot of grounds" and their intersections contain the reducible elements? I do not know whether my visualization is correct  .

In a group, normal subgroups are most likely not cyclic and hence not generated by a single element. Ideals, as kernels of homomorphisms, is a similar concept to normal subgroups. Why would we expect some integral domains to have the property which all ideals are principle (generated by one single element)?

As a second part of the question, what is the best way to visualize principle ideal domains in general? Since prime ideals are maximal, these ideals "cover a lot of grounds" and their intersections contain the reducible elements? I do not know whether my visualization is correct.

In a group, normal subgroups are most likely not cyclic and hence not generated by a single element. Ideals, as kernels of homomorphisms, is a similar concept to normal subgroups. Why would we expect some integral domains to have the property which all ideals are principle (generated by one single element)?

As a second part of the question, what is the best way to visualize principle ideal domains in general? Since prime ideals are maximal, these ideals "cover a lot of grounds" and their intersections contain the reducible elements? I do not know whether my visualization is correct.

In a group, normal subgroups are most likely not cyclic and hence not generated by a single element. Ideals, as kernels of homomorphisms, is a similar concept to normal subgroups. Why would we expect some integral domains to have the property which all ideals are principle (generated by one single element)?

As a second part of the question, what is the best way to visualize principle ideal domains in general? Since prime ideals are maximal, these ideals "cover a lot of grounds" and their intersections contain the reducible elements? I do not know whether my visualization is correct  .