By T.Y. Lam

ISBN-10: 0387975233

ISBN-13: 9780387975238

ISBN-10: 3540975233

ISBN-13: 9783540975236

Through aiming the extent of writing on the beginner instead of the gourmand and through stressing the position of examples and motivation, the writer has produced a textual content that's compatible for a one-semester graduate path or for self-study.

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**Extra resources for A first course in noncommutative ring theory**

**Sample text**

A function with this property is called one-to-one. This means that f is one-to-one if whenever f (a) = f (a ), then a = a . 7. 6 are one-to-one. 8. 6? ” If domains agree and it makes sense to add, subtract, or multiply elements of the range, we can combine two functions to get a new one. Dividing functions may require some additional restrictions, however. Example. Suppose f is the function f (x) = x2 and g is the function g(x) = x + 1, where the domain of f and g is R. Then we can create a new function (f + g)(x) by adding f and g.

There is another way to combine functions that is useful. Definition. Suppose A, B, and C are sets and f and g are functions, where f : A → B and g : B → C. The composition of f and g is a function g ◦ f : A → C defined by (g ◦ f )(a) = g(f (a)). Example. Suppose f is the function f (x) = x2 and g is the function g(x) = x + 1, both with domain R. Then (g ◦ f )(x) = g(x2 ) = x2 + 1. The composition (f ◦ g)(x) = f (x + 1) = (x + 1)2 . Both g ◦ f and f ◦ g also have domain R. Note that f ◦ g = g ◦ f .

What is A ∩ B? 4. }. What is A ∪ B? What is A ∩ B? 5. Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}. What is A × B? What is B × A? 6. Let A and B be as above. Which of the following represent a function from A into B? F = {(1, 2), (2, 3), (3, 4), (4, 5)}; G = {(1, 4), (2, 4), (3, 3)}; H = {(2, 1), (1, 3), (2, 3), (3, 5)}; I = {(1, 5), (3, 3)}; J = {(1, 6), (2, 7), (3, 8)}; K = {(3, 3), (2, 4), (1, 5)}. 7. If S is a set with n elements, show that the power set of S has 2n elements. 8. The set {(1, 11), (2, 15), (3, 10), (4, 13), (5, 12), (6, 14), (7, 13)} is a function.

### A first course in noncommutative ring theory by T.Y. Lam

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