question:What is a principal ideal domain and how does it relate to other algebraic structures? A principal ideal domain (PID) is an integral domain in which every ideal is principal, meaning it is generated by a single element. In other words, for every ideal I in a PID, there exists an element a such that I = (a), where (a) denotes the set of all multiples of a. One of the key properties of a PID is that it is a unique factorization domain (UFD), meaning that every non-zero element can be factored into a product of prime elements in a unique way, up to units. This is because, in a PID, every ideal is principal, and therefore, every element can be expressed as a product of prime elements. PIDs are closely related to other algebraic structures, such as Euclidean domains. A Euclidean domain is an integral domain that is equipped with a Euclidean function, which is a function that assigns a non-negative integer to each non-zero element of the domain. The Euclidean function satisfies certain properties that allow us to perform the Euclidean algorithm, which is a method for finding the greatest common divisor of two elements. In fact, every Euclidean domain is a PID, because the Euclidean algorithm allows us to find the greatest common divisor of two elements, and this greatest common divisor generates the ideal that contains both elements. Conversely, not every PID is a Euclidean domain, because there are PIDs that do not have a Euclidean function. Another important relation is between PIDs and Dedekind domains. A Dedekind domain is an integral domain that is Noetherian, meaning that every ideal is finitely generated, and integrally closed, meaning that it contains all the roots of monic polynomials with coefficients in the domain. Every PID is a Dedekind domain, but not every Dedekind domain is a PID. PIDs also play an important role in number theory, particularly in the study of prime numbers and modular forms. For example, the ring of integers is a PID, and the prime ideals in this ring correspond to the prime numbers. In addition, PIDs have applications in algebraic geometry, particularly in the study of curves and surfaces. For example, the coordinate ring of a curve or surface is often a PID, and the prime ideals in this ring correspond to the points on the curve or surface. In conclusion, principal ideal domains are important algebraic structures that have many interesting properties and applications. They are closely related to other algebraic structures, such as Euclidean domains and Dedekind domains, and play an important role in number theory and algebraic geometry. One natural question that arises is what are some examples of PIDs? Some examples of PIDs include the ring of integers, the ring of Gaussian integers, and the ring of Eisenstein integers. These rings are all PIDs because they are Euclidean domains, and therefore, every ideal is principal. Another question that arises is what are some properties of PIDs that make them useful in algebra and number theory? Some properties of PIDs that make them useful include the fact that every ideal is principal, the fact that they are unique factorization domains, and the fact that they are Noetherian and integrally closed. In what ways do PIDs relate to other areas of mathematics, such as algebraic geometry and number theory? PIDs have many connections to other areas of mathematics, particularly in the study of curves and surfaces in algebraic geometry, and in the study of prime numbers and modular forms in number theory. Let's consider more questions about these and related topics.

question:If sin frac{alpha}{4} = frac{sqrt{3}}{3}, then cos frac{alpha}{2} = A: -frac{2}{3} B: -frac{1}{3} C: frac{1}{3} D: frac{2}{3}

question:Write a summary based on this article: World's Only Drive-Thru Strip Club (Closed) Congruity, Pennsylvania The drive-thru window at the Climax Gentleman's Club offers the novelty of a drive-thru tree with the I'll-do-it-in-my-car convenience of a drive-thru McDonald's. A special gravel driveway leads to a cinder block carport at the back of the building, where patrons can watch the indoor activity from their vehicles, though a diamond-shaped window. A pay here booth takes credit cards, and displays an autographed photo of Fred The Honzman Honsburger, a right-wing radio talk show host on Pittsburgh's KDKA. Current rates are 20 a minute for two or more people; 10 a minute for solo customers. If there's someone in front of you, you just have to wait in your car. We passed through too early to test out the window (the club was officially closed, though we stepped aside so a customer buying a private dance could enter). The club owner, Nick Climax agreed to speak by phone. We're the first one in world history, says Climax of his drive-thru. It was built in the spring of 2000, and remains the only one that he knows of. The media pounced on the story at the time, and Nick proudly reels off his list of conquests: Jay Leno, Howard Stern, Penthouse, Esquire. The drive-thru became a question in the 20th anniversary edition of Trivial Pursuit. More people listened to my radio interview in Australia in 2000 than listened to the Olympics, Nick tells us. I turned down the cover of Time magazine on October 2, 2000; I didn't want my picture taken. Thinking about the frustration of a line at the drive-thru, we ask Nick, Is there a best time to come? He replies, misinterpreting the question, Any time we're open is good. ||||| They may look like McDonald's drive-thrus, but motorists pulling up to these windows aren't buying hamburgers or french fries. They're buying booze! And it's legal! Hard to believe, but in states like Louisiana and Texas, drive-thru stands specialize in selling daiquiris and other alcoholic drinks to drivers. Inside the drive-thrus are rows of machines that look like they're dispensing slushees. But they're actually pumping out pre-mixed alcoholic concoctions in all flavors like electric lemonade, pina coladas, jungle juice, and margaritas. They even sell shots of liquor. And surprisingly it's all perfectly legal for drivers to purchase these cocktails as long as they don't drink them while behind the wheel. In one weekend in New Orleans, we watched dozens of drivers making daiquiri pit-stops and then driving off sipping their cocktails from a straw. We spotted one couple grabbing afternoon cocktails filled with vodka at a daiquiri drive-thru. We followed the driver and watched him sip the drink while stopped at a red light. After watching him drive erratically at over 70MPH on the freeway, INSIDE EDITION'S Chief Investigative Correspondent, Lisa Guerrero, had a few questions for the driver when he finally reached his destination. Are you comfortable with drinking and driving sir? asked Guerrero. The driver didn't respond. INSIDE EDITION also witnessed another driver who didn't even make it out of the parking lot before he started drinking a daiquiri he ordered with an extra shot of Bacardi. We followed him through one busy intersection after another while he kept sipping his drink. Unbelievably this driver's final destination was a nearby playground. He had a child riding in the back seat the entire time. And even though we heard him order an alcoholic daiquiri, the driver told us it was non-alcoholic. It's a virgin, said the driver. Really, responded Guerrero, you're trying to tell me that's a virgin daiquiri? The driver then got back in his car and prepared to take off. Excuse me, yelled Guerrero at the woman who was putting the child back in the car. Are you sure your child should be in that car? He's been drinking and your putting a child in that car. The owners of these drive-thru bars say they abide by the law and insist they don't encourage people to drink and drive. But that's exactly what we found customers doing. We had a few questions for one of the managers at the daiquiri stands but he wasn't happy to see us. You're serving alcohol to people in their cars. Don't you think this is dangerous? asked Guerrero. It's legal, replied the manager before walking away. We showed our video to Laura Dean-Mooney, the president of Mothers Against Drunk Driving. When you allow alcohol to be delivered through a window of a vehicle, you're opening the door for a disaster, said Dean-Mooney. ||||| From strip clubs to funerals, there seems to be a drive-thru available for just about everything in this great, car-obsessed nation. CC BY: Kevin Spencer We Americans sure love our cars — and the convenience of going about our business without ever having to leave the driver's seat. Here, five unusual drive-thru options from around the country: 1. Funerals The Robert L. Adams Mortuary in Compton, Calif., is one of a handful of funeral parlors in the U.S. to offer drive-thru viewings of the dead. You can come by after work, you don't need to deal with parking, you can sign the book outside, and the family knows that you paid your respects, says owner Peggy Scott Adams, as quoted in the Los Angeles Times. It's a convenience thing. 2. Strip clubs The claim to fame of the Climax Gentlemen's Club in Delmont, Pa., was its diamond-shaped drive-thru window. The club, apparently closed since 2009, allowed motorists to pay 10 to view strippers through a small window, without having to leave their cars. Proponents tout the safety and convenience of the drive-thru window, said Gamaliel in a 2000 post at everything2.com. But what about the loss of that lovely strip club ambiance? 3. Prayers The Cathedral of Faith Church of God in Christ in Mount Morris Township, Mich., has held drive-thru prayer services, according to USA Today. Those who heeded signs that read Pull over for Prayer! could fill out a form describing what they wanted to pray about, and then steer over to a minister who would pray with them while their cars idled. Taking the church to the people, to the streets, is what this is all about, says the Rev. Chris Martin. 4. Daiquiris Despite the nationwide crackdown on drunk driving, motorists in states like Louisiana and Texas can still find drive-thru frozen daiquiri chains. And the boozy pit stops are perfectly legal as long as the servers put tape across the straw hole on the lid, and hand you a straw separately. Gallon-sized drinks are available in some locations. For anyone mildly interested in breaking the law, they certainly make things easy, says Tim Murphy at Mother Jones. 5. Liquor... and guns Why stop just for liquor when you can get a gun, too? The Double Shot liquor and gun store in Schulenburg, Texas, has every liquor brand and firearm brand, according to its Yahoo! directory listing. Only in Texas can you buy a bottle of whisky and a gun at the same time, says Rastyrori at the Los Angeles Times — and do it from your car. The store has the convenience of drive-thru so you don't have to get out from behind the wheel while you drink and shoot. |||||

question:Find the longest common prefix among the following list of strings: ['rhinoceros', 'rhino', 'rhubarb', 'rhapsody']: