Ring Theory Quiz

The ideal generated by a set $S$ in a ring $R$ is the smallest ideal that contains all elements of $S$. It is denoted as $\langle S \rangle$.

A quadratic integer ring is a ring that consists of all numbers of the form $a + b\sqrt{d}$, where $a$ and $b$ are integers and $d$ is a square-free integer.

An example of a non-commutative ring is the ring of 2x2 matrices with real entries under matrix addition and multiplication.

<p>A trivial ring is a ring that contains only one element, which is the additive identity. It is called a zero divisor because multiplying any element in the ring by the additive identity results in the additive identity.</p> Signup and view all the answers

<p>A maximal ideal in a ring is an ideal that is not properly contained in any other proper ideal of the ring. In other words, if $I$ is a maximal ideal in a ring $R$, then there is no ideal $J$ such that $I \subsetneq J \subsetneq R$.</p> Signup and view all the answers

<p>Units in a ring are elements that have a multiplicative inverse. In other words, for an element $a$ to be a unit in a ring, there must exist an element $b$ such that $ab = ba = 1$.</p> Signup and view all the answers

<p>The concept of zero divisor is important in ring theory because it helps to characterize the properties and structure of rings. Rings without zero divisors have additional algebraic properties and are used in various areas of mathematics and physics.</p> Signup and view all the answers

<p>A zero divisor in a ring is a non-zero element that, when multiplied by another non-zero element, results in the additive identity. In the trivial ring, every non-zero element is a zero divisor because multiplying it by the additive identity gives the additive identity.</p> Signup and view all the answers