Suppose a company's name is trademarked but their product name is NOT. In that case, is it good to register domain name on the product name?
The problem with this form of question is the fundamental misunderstanding contained within it, signified by the use of "trademark" as a verb.
It is an unfortunate fact of life that domainers in general continue to propagate the common misconception that "having a trademark" is the same thing as "registering a trademark", and that they consequently tend to believe that there is no trademark absent registration.
Whether their product name is, or is not, a trademark, depends on whether it is distinctive of the origin of that product.
For example, if Whatsit Breweries sells a product made of barley mash, hops, yeast and water and calls it "beer", then they don't have a trademark in the name of that product, because "beer" is not distinctive of any one maker of beer. It is the generic term for beer, and thus cannot function as a trademark.
If, on the other hand, Whatsit Breweries sells the same product and calls it "My Dog's Balls",
and if nobody else is selling beer called "My Dog's Balls",
and if "My Dog's Balls" is not a famous mark for something other than beer,
and if consumers recognize "My Dog's Balls" as signifying a brand of beer such that they can distinguish it from other brands of beer on a shelf or at the tap...
...then, in the US, UK, and any other common law jurisdiction in which they sell that beer, Whatsit Breweries bloody well
has a trademark in "My Dog's Balls" for beer.
That is what it takes to "get a trademark" in the US, UK, Canada, Australia, and a bunch of other significant jurisdictions. They don't have to file an application with some government office. They don't have to be in some database of pending, failed and registered applications. They don't have to get a fancy piece of paper from the government.
THEY HAVE A TRADEMARK.
That's the first thing you need to understand. You need to get past this idea that registered trademarks represent the sum total of trademarks.
IF they have a trademark,
THEN they might want to apply to register that trademark, since registration provides a number of legal advantages if they want to enforce their mark. A registration, in the US, for example, provides them with presumptive validity, nationwide effect, presumed notice, and an number of other conveniences relative to a common law trademark.
Okay, so, if you have grasped that initial important concept, we can move to the next part of the question:
Someone comes along and registers MyDogsBalls.com. Is that okay?
Well, we don't have enough information to say whether it is okay or not.
What are they doing with that domain name?
Let's say the corresponding website has information on where you can get your pet neutered. Maybe it is advertising a veterinary office who has a special price for neutering pets.
Maybe the website has ads for pet toys, such as balls that your dog likes to play with.
Those kinds of things would be a straightforward use of MyDogsBalls.com that wouldn't appear to have anything to do with Whatsit's trademark for beer. Those kinds of things would apparently be making use of the directly denotative sense of the phrase "my dogs balls" whether referring to reproductive organs or to toys.
If, on the other hand, the domain name is parked with ads for beer then, yeah, there's a real problem there. Obviously, if the way the domain name is being used has something to do with the trademark sense of the phrase, which is distinctive of Whatsit's beer, then one could reasonably conclude the domain name was registered to take advantage of the goodwill signified by their mark in beer.
But, let's say that the website isn't doing anything that seems to do with beer or with dogs' balls. In that case, there might be other evidence - what sort of other domain names does this person have, or what sort of behavior in which this person has engaged - which might provide some insight into this person's motivation.
Finally, there is the notion of "fame" in trademark law, which is important. Take "Coca-Cola" for example. The words "coca" and "cola" merely refer to two names of plants which are used in its production. The thing is, it is one of the most well-known trademarks on the planet. So, if you are making pants, and calling them "Coca-Cola" you are not going to have a decent argument of "I'm not infringing their mark" because Coca-Cola is so well-known that pants marked "Coca-Cola" might reasonably be understood to be authorized or licensed by them, regardless of what, specifically, their mark is for.
Okay, so, if you are with me on the basics of "what is a trademark in the first place and how do you get one" and the importance of "what other evidence might there be of intent", you can move on to the other question above: timing.
If you registered the domain name well before Whatsit ever started making "My Dogs Balls" beer, then obviously your registration of the domain name could not have been motivated by their mark. That's the easy case.
But if they started making their beer in January 2010 and then later proceeded to register their mark in December 2015, and you registered your domain name in, say, June 2012, then there are going to be some questions. How well known was their beer in 2012? Is it likely you were aware of their beer? And, again, what were you doing with the domain name?
There are a number of other edge cases that can be built on this scenario. Let's say they were a small regional beer in Oregon in June 2012, and you live in Florida. Also, let's say you have 100 domain names and 75 of them are pet-related and none of them are beer related. You could easily make the case that your registration of the domain name didn't have anything to do with their mark for beer, and was consistent with your other pet-related domain names.
But, let's say that instead, you have 100 domain names, and 75 of them are also the names of craft beers from all over the country, and none of your domain names have anything to do with pets. Well, then it is flipping obvious that your registration of the domain name was motivated by their mark for beer.
Also, maybe they were originally a small craft brewery somewhere, but on June 10, 2012, they received national attention in the news or some celebrity endorsed their beer. Then, let's say your domain name was registered on June 11, 2012. What do you think a panel might conclude about why you registered the domain name?
Another common edge case is this one.... Let's say that you registered the domain name in November 2009 - two months before they started selling their beer. In most cases, that would be fine. But, let's add the further facts that, in October 2009, the guy who runs Whatsit Brewery, and was planning to launch "My Dogs Balls" beer broke up with his girlfriend. Then, in November, you started dating her and you happened to register the domain name. What do you think a panel might conclude on those facts?
Here's the problem, most of the questions that get posted here (a) miss the boat on what is a trademark and how do you get one, and then (b) don't include enough factual context to reach any sort of definite conclusion as to how things might work out.
So, getting back to this:
But suppose a company's name is trademarked but their product name is NOT. In that case, is it good to register domain name on the product name?
...the only answer is that it depends on facts other than those stated in the question.