In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which contains the multiplicative identity of R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

The Diagonal Subring and the Cohen-Macaulay Property of a Multigraded Ring
Each countable reduced torsion-free commutative ring is a pure subring of an e-ring
AN EQUIVALENCE BETWEEN RING F AND INFINITE MATRIX SUBRING OVER F
On Rees Algebras with a Gorenstein Veronese Subring ☆
Non-Noetherian rings for which each proper subring is Noetherian
On the subring structure of finite nilpotent rings
Germs of arcs on singular algebraic varieties and motivic integration
Dynamical Systems of Algebraic Origin
Secure execution of program code
Roughness in rings