MTH2001 Algebraic Structures Flashcards

Front: What is a ring? Back: A set $R$ equipped with two binary operations (addition and multiplication) such that ((R, +)) is an abelian group, multiplication is associative, and multiplication distributes over addition. If there exists a multiplicative identity, it is called a ring with unity.

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Front: What are examples of rings? Back: $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ (reals), $\mathbb{C}$ (complexes), and polynomial rings like $R[x]$.

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Front: What is a polynomial ring? Back: The set of all polynomials with coefficients in a ring $R$, denoted $R[x]$, equipped with standard polynomial addition and multiplication.

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Front: What is a subring? Back: A subset $S$ of a ring $R$ that is itself a ring under the same operations as $R$.

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Front: State the Quick Subring Theorem.

Back: A non-empty subset $S$ of a ring $R$ is a subring if $S$ is closed under subtraction ($a-b\in S$) and multiplication ($ab\in S$) for all $a,b\in S$.

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Front: What is a zero divisor in a ring?

Back: Non-zero elements $a,b\in R$ such that $ab=0$.

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Front: What is an integral domain?

Back: A commutative ring with unity that has no zero divisors (i.e., if $ab=0$, then $a=0$ or $b=0$).

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Front: What are the cancellation laws in an integral domain?

Back: If $a\ne 0$ and $ab=ac$, then $b=c$. Cancellation is valid due to the absence of zero divisors.

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