Billy Bob said:Is it or isn't it? Usually you would try to show it isn't, which you did, without success, so maybe it is. Another clue: what are the easiest examples (in your book or notes) of integral domains that are not PID's? Are they more complicated than Z[1/5]? I bet they are. So that's a clue that this one is.
Try to show it is. I would just try to show it directly. Two hints. First, what do elements of Z[1/5] look like? Second, how do you even show (directly) that Z is a PID?
A PID (Proportional-Integral-Derivative) controller is a feedback control system commonly used in engineering and science. It uses a control algorithm to adjust a process variable based on the difference between a desired setpoint and the current value of the variable.
In mathematics, Z[1/5] refers to the set of rational numbers with a denominator of 5. A PID in this context means that this set satisfies the properties of a Principal Ideal Domain, which is a type of algebraic structure used in abstract algebra.
A PID must have the following properties: 1) It is an integral domain, meaning that it has no zero divisors; 2) It is a principal ideal ring, meaning that every ideal in the ring can be generated by a single element; 3) It is a unique factorization domain, meaning that every element can be factored into a unique product of irreducible elements.
To determine if Z[1/5] is a PID, we must check if it satisfies the three properties mentioned above. In this case, Z[1/5] is indeed a PID because it is an integral domain, a principal ideal ring, and a unique factorization domain.
PID structures have many applications in mathematics and engineering, such as in control systems, signal processing, and coding theory. Having a set like Z[1/5] that satisfies the properties of a PID allows for the use of these structures in solving problems and proving theorems in these fields.