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Let $F$ be a field and $f(x) \in F[x]$ be a polynomial of degree $> 1$. If $f(a) = 0$ for some $a \in F$, then $f(x)$ is reducible over $F$.

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#abstract-algebra #ring-theory
166
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How is a dimension of an ideal defined in $R$

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#abstract-algebra #ring-theory
49
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Isomorphic products of quotient rings

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#abstract-algebra #proof-verification #ring-theory
144
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How to find the extension of an ideal

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#abstract-algebra #ring-theory #ideals
116
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Rings in which each element is a sum of $n$ commuting idempotents

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#abstract-algebra #ring-theory #noncommutative-algebra #positive-characteristic #boolean-ring
226
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Alternative definition on free modules.

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#abstract-algebra #ring-theory #modules #free-modules #free-abelian-group
215
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Universal Property of the Ring of Quotients.

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#abstract-algebra #ring-theory #universal-property
56
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Let $A$ be any commutative ring and consider $\operatorname{Spec}(A)$, then for any $E \subseteq A$ we have $V(E) = \emptyset \iff E = A$

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#ring-theory #commutative-algebra
195
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Morphism between quotients of a ring by ideals.

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#abstract-algebra #ring-theory
179
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$(ab)^n = a^{n}b^{n}$ in a unitary ring $A$ implies A commutative?

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#abstract-algebra #ring-theory
151
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Ring with infinitely many totally ordered prime ideals

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#abstract-algebra #algebraic-geometry #ring-theory #maximal-and-prime-ideals
33
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Showing that three rings are isomorphic

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#ring-theory #group-rings #ring-isomorphism
658
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Why are ideals specifically additive subgroups?

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#abstract-algebra #ring-theory #ideals
152
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Is it true that $A \otimes_{\Bbb Z} B= A \otimes _{ R} B$ for every commutative ring $R$?

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#abstract-algebra #ring-theory #tensor-products
65
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Centre of algebra under field extension

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#abstract-algebra #ring-theory #extension-field

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