Euclidean rings are a type of commutative ring that have a division algorithm. In other words, given any two elements a and b in a Euclidean ring, with b not equal to zero, there exist elements q and r such that a = bq + r, where r is either zero or has a “smaller” norm than b.

The norm function in a Euclidean ring is a function that assigns a non-negative integer to each non-zero element of the ring. This norm function is required to satisfy certain properties, such as the norm of the sum of two elements being less than or equal to the maximum of their individual norms.

The division algorithm in a Euclidean ring allows for the division of elements, similar to how division works in the integers. This property makes Euclidean rings useful in various areas of mathematics, such as number theory and algebraic geometry.

Examples of Euclidean rings include the integers, the Gaussian integers (complex numbers of the form a + bi, where a and b are integers), and the polynomials with integer coefficients.