Algebra 1: Groups, Rings, Fields and Arithmetic (Lal)

ramji-lal (HRI Allahabad), Springer 2017, in the infosys-science-foundation-series (Mathematical Sciences sub-series). Eleven chapters over ~433 pages, DOI 10.1007/978-981-10-4253-9. First of three volumes; the publisher’s description calls it “the first in a series of three volumes dealing with important topics in algebra.”

This closes the spoke’s abstract algebra and number theory growth edge — the last untouched subject on the list and, as that edge put it, “the next structural layer under linear-algebra.”

What this page rests on. The publisher’s book page: description, chapter list, pagination, series and price. The body was not read — the copy that reached the hub is a scanned PDF on a file-sharing link, and the publisher’s own metadata is both the better citation and sufficient for what this page claims. No statement below is a claim about the mathematics inside; they are claims about the book’s scope and position.

Contents

The order is the notable part:

1. Language of Mathematics 1 (Logic) — 12 pages
2. Language of Mathematics 2 (Set Theory) — 41 pages
3. Number System — 37 pages
4. Group Theory · 5. Fundamental Theorems · 6. Permutation Groups and Classical Groups — the group-theory core, ~125 pages
5. Elementary Theory of Rings and Fields — 50 pages
6. Number Theory 2 — 41 pages
7. Structure Theory of Groups · 10. Structure Theory Continued — ~75 pages
8. Arithmetic in Rings — 33 pages

Two chapters of logic and set theory before any algebra, then number systems, then groups. The foundations are built rather than assumed, and the number-theoretic thread is woven through the algebraic one instead of being separated from it — arithmetic returns in chapter 8 and again in chapter 11, once rings are available to say it properly.

What each chapter says it does (from the publisher’s public chapter abstracts, 2026-08-08):

• Logic — a foundation for “sound mathematical reasoning” and for what a proof is, motivated by Russell’s and Cantor’s paradoxes and the turn-of-the-century demand for rigorous foundations; “in logic, the interest is in the form rather than the content of the statements.”
• Set theory — names the two axiomatic systems and picks one: “We follow the Zermelo–Fraenkel axiomatic system together with the axiom of choice,” over Gödel–Bernays.
• Number System — the real and complex systems developed “starting from Peano’s axiom”, plus linear Diophantine equations and linear congruences.
• Group Theory — the basic introduction. Fundamental Theorems — Lagrange, the isomorphism theorems, and direct decomposition into indecomposable groups. Permutation and Classical Groups — “the two main sources of groups are the permutation groups and the matrix groups.”
• Rings and Fields — a structure “whose intrinsic presence in almost every discipline of mathematics is frequently noticed”, developed deliberately “on the pattern the theory of groups was developed.”
• Number Theory 2 — arithmetic functions, quadratic residues, Gauss’s quadratic reciprocity law, nth-power residues.
• Structure Theory of Groups (two chapters) — “the main problem in the theory of groups is to classify them”: the Sylow theorems, classification of finite abelian groups, Schreier and Jordan–Hölder, then Remak–Krull–Schmidt, solvable and nilpotent groups, and presentation theory.
• Arithmetic in Rings — rings “in relation to their arithmetical properties.”

Those abstracts are the publisher’s public metadata, not the chapters. They fix the scope precisely and say nothing about the exposition.

That places the book on this spoke’s foundations question from a new side. The corpus’s argument so far has been about which foundation a first course should stand on — Keisler replacing limits with infinitesimals, Stirzaker defending the formal definition on pedagogical grounds. Lal does not argue for a foundation; he teaches it first, as chapters, before the subject starts. Whether that works is not something the table of contents can settle.

Audience

“Intended as a text for undergraduate and graduate students of mathematics” — a wider span than any other text here claims, and a caution when reading anything sourced from it: which level a given chapter is pitched at is not visible from outside.

The access axis, broken again

Every free text in this corpus is free in a different way — hefferon-linear-algebra and trench-real-analysis under real free licences, herzog-understanding-statistics open access but CC BY-NC, corral-vector-calculus self-published, mcmullen-probability-theory course notes. This one is commercial: $49.99 for the eBook, $89.99 in hardcover, no open licence.

The pattern is now visible twice. Statistics arrived first as a paid text (larsen-marx-mathematical-statistics, Pearson) and an open one followed two days later; algebra arrives paid. The subject gets covered before it gets covered freely, and a corpus that only accepted open licences would have neither subject today.

Its free counterpart, same day

judson-abstract-algebra was pulled in hours later to close the licence half of the edge this book closed the subject half of. The two make abstract-algebra the corpus’s first subject with two treatments that disagree about what a first course is for — foundations-first and classification-heavy here, applications-forward and Galois-complete there. Where this page describes that contrast, it is comparing stated scope: Judson’s body has been read, Lal’s has not.

Connections

• author ramji-lal · publisher springer · series infosys-science-foundation-series · affiliation harish-chandra-research-institute
• closes the abstract-algebra edge; the structure under linear-algebra; six of eleven chapters are group-theory
• shares a publisher with herzog-understanding-statistics and an access position with larsen-marx-mathematical-statistics; the free counterpart is judson-abstract-algebra