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		{"id":"2e8fdfdf21952821","x":-202,"y":80,"width":250,"height":160,"type":"text","text":"Integral domain\n*  no zero divisors\n* commutative\n* has 1"},
		{"id":"ee7162948cd18471","x":-202,"y":-235,"width":250,"height":151,"type":"text","text":"Division ring\n* every nonzero *x* has an inverse\n* *field* if commutative"},
		{"id":"547d48b7f92b6e4b","x":-1480,"y":-250,"width":555,"height":360,"type":"text","text":"Simple properties:\n1. 0a = a0 = 0\n2. (-a)b = a(-b) = -(ab)\n3. (-a)(-b) = ab\n4. R has 1 => 1 is unique and -a = (-1)a\n   \nAdvanced properties:\n\n5. *not* zero divisor => cancellation property \n6. Every nonzero element of commutative ring that is not a zero divisor has a mult-inverse in some larger ring\n7. "},
		{"id":"b4beb04a23e91013","x":260,"y":-40,"width":347,"height":200,"type":"text","text":"For nonzero **p(x)** and **q(x)** in **R\\[x\\]**  (polynomial with \"integral\" coefs):\n1. deg pq = deg p + deg q\n2. units of R\\[x\\] are units of R only\n3. R\\[x\\] is integral domain"},
		{"id":"e3c87288e77e578d","x":260,"y":210,"width":250,"height":60,"type":"text","text":"Any finite integral domain is a field"},
		{"id":"b3aaf4b0fe583c23","x":260,"y":320,"width":361,"height":129,"type":"text","text":"Cancellation property for any a, b, c:\n* ab = ac => a = 0 or b = c "},
		{"id":"06ddf268d5639bbd","x":184,"y":-189,"width":250,"height":60,"type":"text","text":"Every finite division ring is commutative, i.e. a field"},
		{"id":"5060f6ebe8169488","x":-277,"y":500,"width":400,"height":177,"type":"text","text":"Notable examples:\n1. Quadratic field Q\\[D\\]\n$$\na + b \\sqrt{D} \\:|\\: a,b \\in Q  \\: and \\: D \\: is \\: squarefree \\: integer\n$$\n2.\n"},
		{"type":"text","text":"Ring\n1. abelian for +\n2. associative for *\n3. distributive from left and right *","id":"2f5ef207faf452e3","x":-734,"y":-159,"width":220,"height":180},
		{"id":"f8294ff4f995bde9","x":-749,"y":160,"width":250,"height":123,"type":"text","text":"Subring:\n1. Subgroup under +\n2. Closed under *"},
		{"id":"9173bc44a7ce8d6d","x":-840,"y":389,"width":432,"height":400,"type":"text","text":"Notable examples:\n1. $$\nZ[\\sqrt{D}]\n $$\n2. $$\nZ[(1 + \\sqrt{D})/2]\n $$ \n when D = 1 mod 4\n \n \n 3. Ring of integers inside quadratic field\n      *O* = Z\\[w\\] = {a + bw}\n      w = sqrt(D)   when    D = 2,3 mod 4\n      w = 1 + sqrt(D)/2  when D = 1 mod 4\n \n"},
		{"id":"4b3393d7d80bd0d6","x":-1442,"y":351,"width":250,"height":109,"type":"text","text":"Ring homomorphism:\n1. f(a + b)  = f(a) + f b)\n2. f(ab) = f(a)f(b)"},
		{"id":"771ce7691ba0f114","x":-1442,"y":540,"width":250,"height":98,"type":"text","text":"1. Ker f is a subring\n2. Im f is a subring\n3. a \\* Ker f = Ker f"},
		{"id":"03d942be3208b083","x":-1452,"y":740,"width":250,"height":165,"type":"text","text":"Quotient ring f/Ker f:\n* fibers have a ring structure\n* isomorphic to Im f"},
		{"id":"da006d3ea86dca99","x":-1480,"y":1120,"width":250,"height":166,"type":"text","text":"Notion of ideal:\n*  subring which is closed under left/right/both multiplication"}
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