definitions in algebra I am interested in algebra and I want to read some good books about it. I have a problem with some definitions like free group or algebra. In some books there are different definitions of them and I'm confused. I want to read a reference book which has exact and specific definitions and main theorems in it.
 A: Study Dummit & Foote. It has what you may call exact definitions and all. Sometimes it gives multiple definitions but explains how they are equivalent.
A: I claim there is no true, objective difference between a property and a definition. Part of creating mathematics is deciding what are properties that need to be proven from definitions versus what are instead defining properties. Often there are morally correct definitions: those that get the greatest traction in the begining, or the greatest mileage in the end, or are the most aesthetically elegant, or are the most suggestive and illuminating as to their purpose and discovery, or mimic our intuition closest, or have other desirable properties. I am a moral relativist, though, believing that which definition of something is best (even "morally correct") can depend on context.
On the one hand, it would be nice if every concept had a single definition, and every other property was derivative - this would make learning much more straightforward. But when there is no universally best definition (just look up "Prufer domain"), it is a disservice to other definitions to pick just one for everybody; furthermore, the ability to convert between definitions is indicative of a fuller understanding of all of the properties related to a given concept. Indeed, being able to think about a concept from multiple points of view is extremely useful in practice - then all of the basic properties feel "defining" and are more memorable. Showing that different definitions are equivalent, and then being able to feel that their equivalence is "obvious," is a rite of passage.
Thus I am forced to reject the premise of this question. Being exposed to multiple definitions of things is good for learning math, and having multiple definitions is healthy for the whole enterprise of doing math. In any case, no book of definitions can change the fact that multiple definitions exist out there in the world and are in wide usage by different people and sources.
You might, though, find some definitions hard to understand, or hard to see the motivation of, or hard to see how to use, at least at first. To this end I can recommend shopping: shop for the "right" definitions of concepts for you, the ones that are the most accessible or intuitive. I do this all the time, which is why I am rarely satisfied with reading just one source (I go shopping for more than just definitions, though). Your preferences may differ from that of others, but a good heuristic is that the more basic or introductory texts will have definitions that are easier to get a foothold with, and you want to learn these first.
For abstract algebra, the text I used in my educational track was Gallian's Contemporary Abstract Algebra, covering groups, rings and fields, which was chosen for simplicity. I don't think it covers algebras (the ones over scalars) or free groups. There are a number of good sources online that can be found through googling, though. The shopping you'll have to do on your own, or ask as a separate question (although asking for introductory texts has been done enough times on MSE).
A: Algebra by T. Hungerford is a very good book and you may find it very useful. 
