Contemporary Abstract Algebra
Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Concept explainers

Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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Chapter 12, Problem 15E

Show that if m and n are integers and a and b are elements from aring, then ( m ⋅ a ) ( n ⋅ b ) = ( m n ) ⋅ ( a b ) . (This exercise is referred to inChapters 13 and 15.)

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Chapter 12 Solutions

Contemporary Abstract Algebra

Ch. 12 - The ring {0, 2, 4, 6, 8} under addition and...Ch. 12 - Find an integer n that shows that the rings Zn...Ch. 12 - Show that a ring is commutative if it has the...Ch. 12 - Prove that the intersection of any collection of...Ch. 12 - Let a, b, and c be elements of a commutative ring,...Ch. 12 - Let a andb belong to a ring R and let mbe an...Ch. 12 - Show that if m and n are integers and a and b are...Ch. 12 - Show that a ring that is cyclic under addition is...Ch. 12 - Let R be a ring. The center of R is the set...Ch. 12 - Let R be a commutative ring with unity and let...
Ch. 12 - Show that a unit of a ring divides every element...Ch. 12 - Suppose that R is a ring with unity 1 and a is an...Ch. 12 - Suppose that there is a positive even integer n...Ch. 12 - Prob. 45ECh. 12 - Show that 2Z3Z is not a subring of Z.
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