TITLE: Non-Commutative Algebra book
QUESTION [9 upvotes]: I am currently having a Master in topics such as Algebra, Geometry and Number Theory and recently I started studying Representation Theory where I've seen definition, theorems and propositions concerning rings that are non-commutative in general. I am wondering if there's any "standard" book(s) that can introduce me to the theory of non-commutative algebra.

REPLY [7 votes]: Ones I have used extensively

First Course in Noncommutative Rings by T.Y. Lam
Isaacs Graduate algebra
Basic abstract algebra by P. B. Bhattacharya, ‎S. K. Jain & Nagpaul

Others often mentioned as standard (but I did not really use)

McCoy's The theory of rings
Dummit and Foote's Abstract algebra
Hungerford's Algebra
Noncommutative rings by Herstein

Ones I have not used extensively but would recommend

Lambek's Lectures on rings and modules
Jacobson's Basic Algebra volumes I and II

Books that are a step up from an introduction

Anderson and Fuller's Rings and categories of modules
Lam's Lectures on modules and rings
Louis Rowen's books and Carl Faith's books on algebra and ring theory.