By David R. Finston and Patrick J. Morandi

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A matrix A is in row reduced echelon form if 1. the …rst nonzero entry of any row is 1. This entry is called a leading 1; 2. If a column contains a leading 1, then all other entries of the column are 0; 3. If i > j, and if row i and row j each contain a leading 1, then the column containing the leading 1 of row i is further to the right than the column containing the leading 1 of row j. To help understand Condition 3 of the de…nition, the leading 1’s go to the right as you go from top to bottom in the matrix, so that the matrix is in some sense triangular.

In a linear algebra course you saw that arithmetic operations can be de…ned on matrices, and that there are structures called vector spaces in which you can add and subtract vectors and perform scalar multiplication. In calculus, if not earlier, you saw that functions can be combined by adding, subtracting, multiplying, and dividing, along with composing them. There are many similarities between these di¤erent systems. The idea of abstract algebra is to distill down the ideas and properties common to these systems, and then study any system with the same properties.

Z are the elements of the form (a; 0) or (0; a) for 13. Give an example of a ring R and an element a 2 R that is neither a zero divisor nor a unit. 14. Let R be a ring in which 0 = 1. Prove that R = f0g. 15. Prove that the multiplicative identity of a ring is unique. 16. If a and b are elements of a ring R, prove that ( a) ( b) = ab. 17. Prove that if a and b are elements of a ring, then (a + b)2 = a2 + ab + ba + b2 . 18. Let R be a ring in which a2 = a for all a 2 R. Prove that is commutative.

### An Introduction to Abstract Algebra via Applications by David R. Finston and Patrick J. Morandi

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