Definition:Subring
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Definition
Let $\struct {R, +, \circ}$ be an algebraic structure with two operations.
A subring of $\struct {R, +, \circ}$ is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a ring.
Proper Subring
A subring $S$ of $R$ is a proper subring of $R$ if and only if $S$ is neither the null ring nor $R$ itself.
Also defined as
Sources which deal only with rings with unity typically demand that the unity is also part of a subring.
Some sources insist that $R$ must be a ring for $S$ to be definable as a subring, but this limitation is unnecessarily restricting.
Also see
- Results about subrings can be found here.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 19$. Subrings
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: $2.1$ Definition
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 56$. Subrings and Subfields: Definition $1$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): subring
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subring