[Minor observation] Notation: Let ${ (x _j) _{j \geq 1} }$ be a sequence in a normed space. A sequence ${ (x _j) _{ j \in J} }$ with ${ J \subseteq \mathbb{Z} _{\gt 0} }$ infinite is called a subsequence. We say subsequence ${ (x _j) _{j \in J} }$ converges to ${ p }$ as ${ j \in J, j \to \infty }$ if for every ${ \epsilon \gt 0 },$ ${ x _j ... Read more 11 Nov 2021 - 3 minute read
[An old proof. “By finding scaled versions of the triangle within itself”. Yet somewhat mysterious.] So ${c = ( \frac{b}{c} ) b + ( \frac{a}{c} ) a. }$ Read more 09 Oct 2021 - less than 1 minute read
Instructor: Prof. A. V. Jayanthan Link to lectures Book: Atiyah-Macdonald ROUGH NOTES Lec-1: Rings, subrings, homomorphisms, ideals Lec-2: Prime and maximal ideals Lec-3: Every nonzero ring has a maximal ideal. (Apply this to $A/I$ where ideal $I \subsetneq A.$ From correspondence thm, there is a maximal ideal containing $I$. Especially ev... Read more 06 Oct 2021 - 20 minute read
Link to part-1 ROUGH NOTES Lec-21: An ascending chain $I _1 \subseteq I _2 \subseteq \ldots$ of ideals stabilizes if $I _m = I _{m+1} = \ldots$ for some $m$; Ring $R$ is Noetherian (ie every ideal is finitely generated) if and only if every ascending chain of ideals stabilizes $\implies$ Let ideals $I _1 \subseteq I _2 \subseteq \ldots.$ No... Read more 01 Oct 2021 - 13 minute read
Instructor: Prof. Krishna Hanumanthu Link to lectures ROUGH NOTES (!) Lec-1: Rings (Commutative rings with unity. So no to matrix rings). Eg : $\mathbb{Z}, \mathbb{Q}, \mathbb{Z}[i] := \lbrace x + iy : x,y \in \mathbb{Z} \rbrace$ are rings. $\frac{1}{2} \mathbb{Z}$ isn’t, as it has $\frac{1}{2}$ but not $\frac{1}{2} \cdot \frac{1}{2} = \frac{... Read more 25 Sep 2021 - 18 minute read